To find a particular solution to the differential equation using the method of variation of parameters.
we'll follow these steps:
1. Find the complementary solution:
Solve the homogeneous equation x^2y" - 3xy^2 + 3y = 0. This is a Bernoulli equation, and we can make a substitution to transform it into a linear equation.
Let v = y^(1 - 2). Differentiating both sides with respect to x, we have:
v' = (1 - 2)y' / x - 2y / x^2
Substituting y' = (v'x + 2y) / (1 - 2x) into the differential equation, we get:
x^2((v'x + 2y) / (1 - 2x))' - 3x((v'x + 2y) / (1 - 2x))^2 + 3((v'x + 2y) / (1 - 2x)) = 0
Simplifying, we have:
x^2v'' - 3xv' + 3v = 0
This is a linear homogeneous equation with constant coefficients. We can solve it by assuming a solution of the form v = x^r. Substituting this into the equation, we get the characteristic equation:
r(r - 1) - 3r + 3 = 0
r^2 - 4r + 3 = 0
(r - 1)(r - 3) = 0
The roots of the characteristic equation are r = 1 and r = 3. Therefore, the complementary solution is:
y_c(x) = C1x + C2x^3, where C1 and C2 are constants.
2. Find the particular solution:
We assume the particular solution has the form y_p(x) = u1(x)y1(x) + u2(x)y2(x), where y1 and y2 are solutions of the homogeneous equation, and u1 and u2 are functions to be determined.
In this case, y1(x) = x and y2(x) = x^3. We need to find u1(x) and u2(x) to determine the particular solution.
We use the formulas:
u1(x) = -∫(y2(x)f(x)) / (W(y1, y2)(x)) dx
u2(x) = ∫(y1(x)f(x)) / (W(y1, y2)(x)) dx
where f(x) = x^2 ln(x) and W(y1, y2)(x) is the Wronskian of y1 and y2.
Calculating the Wronskian:
W(y1, y2)(x) = |y1 y2' - y1' y2|
= |x(x^3)' - (x^3)(x)'|
= |4x^3 - 3x^3|
= |x^3|
Calculating u1(x):
u1(x) = -∫(x^3 * x^2 ln(x)) / (|x^3|) dx
= -∫(x^5 ln(x)) / (|x^3|) dx
This integral can be evaluated using integration by parts, with u = ln(x) and dv = x^5 / |x^3| dx:
u1(x) = -ln(x) * (x^2 /
2) - ∫((x^2 / 2) * (-5x^4) / (|x^3|)) dx
= -ln(x) * (x^2 / 2) + 5/2 ∫(x^2) dx
= -ln(x) * (x^2 / 2) + 5/2 * (x^3 / 3) + C
Calculating u2(x):
u2(x) = ∫(x * x^2 ln(x)) / (|x^3|) dx
= ∫(x^3 ln(x)) / (|x^3|) dx
This integral can be evaluated using substitution, with u = ln(x) and du = dx / x:
u2(x) = ∫(u^3) du
= u^4 / 4 + C
= (ln(x))^4 / 4 + C
Therefore, the particular solution is:
y_p(x) = u1(x)y1(x) + u2(x)y2(x)
= (-ln(x) * (x^2 / 2) + 5/2 * (x^3 / 3)) * x + ((ln(x))^4 / 4) * x^3
= -x^3 ln(x) / 2 + 5x^3 / 6 + (ln(x))^4 / 4
The general solution of the differential equation is the sum of the complementary solution and the particular solution:
y(x) = y_c(x) + y_p(x)
= C1x + C2x^3 - x^3 ln(x) / 2 + 5x^3 / 6 + (ln(x))^4 / 4
Note that the constant C1 and C2 are determined by the initial conditions or boundary conditions of the specific problem.
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(9).Suppose(r,s) satisfy the equation r+5s=7and-2r-7s=-5 .Find the value of s.
a)-8 b)3 c) 0 d) -1/4 e) none of these (10). Which of the following matrices are orthogonal 20 117 iii) 13-5 iv) 0 02 -1
A rectangular array of characters, numbers, or phrases arranged in rows and columns is known as a matrix. It is a fundamental mathematical idea that is applied in many disciplines, such as physics, mathematics, statistics, and linear algebra.
To solve the system of equations:
r + 5s = 7 ...(1)
-2r - 7s = -5 ...(2)
We can use the method of elimination or substitution. Let's use the method of elimination:
Multiply equation (1) by 2:
2r + 10s = 14 ...(3)
Now, add equation (2) and equation (3) together:
(-2r - 7s) + (2r + 10s) = -5 + 14
3s = 9
s = 9/3
s = 3
Therefore, the value of s is 3.
Answer: b) 3
Regarding the matrices:
i) 20 11
7 -5
ii) 13 -5
-1 2
iii) 0 0
2 -1
iv) 0 0
-1 0
To determine if a matrix is orthogonal, we need to check if its transpose is equal to its inverse.
i) The transpose of the first matrix is:
20 7
11 -5
The inverse of the first matrix does not exist, so it is not orthogonal.
ii) The transpose of the second matrix is:
13 -1
-5 2
The inverse of the second matrix does not exist, so it is not orthogonal.
iii) The transpose of the third matrix is:
0 2
0 -1
The inverse of the third matrix is also:
0 2
0 -1
Since the transpose is equal to its inverse, the third matrix is orthogonal.
iv) The transpose of the fourth matrix is:
0 -1
0 0
The inverse of the fourth matrix does not exist, so it is not orthogonal.
Therefore, the only matrix among the options that is orthogonal is:
iii) 0 2
0 -1
Answer: iii) 0 2
0 -1
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Determine whether the given function is a solution to the given differential equation. 0=4e5t-2 e 2t d²0 de 0- +50= - 7 e 2t dt² dt C d²0 The function 0= 4 e 5t - 2 e 2t a solution to the differential equation de 0 +50= -7 e 2t, because when 4 e 5t - 2 e 2t is substituted for 0, dt² dt equivalent on any intervals of t. de is substituted for and dt is substituted for d²0 d₁² the two sides of the differential equation
The function 0 = 4e^(5t) - 2e^(2t) is a solution to the differential equation d²0/dt² + 50 = -7e^(2t). This is because when the function is substituted into the differential equation, it satisfies the equation for all intervals of t.
To determine whether the given function is a solution to the given differential equation, we substitute the function into the differential equation and check if it satisfies the equation for all values of t.The given differential equation is d²0/dt² + 50 = -7e^(2t). Substituting the function 0 = 4e^(5t) - 2e^(2t) into the differential equation, we have:
d²0/dt² + 50 = -7e^(2t)
Taking the second derivative of the function, we get:
d²0/dt² = (4e^(5t) - 2e^(2t))''
Evaluating the second derivative, we have:
d²0/dt² = (20e^(5t) - 4e^(2t))
Substituting this expression into the differential equation, we have:(20e^(5t) - 4e^(2t)) + 50 = -7e^(2t)
Simplifying the equation, we get:
20e^(5t) + 50 = 3e^(2t)
We can see that this equation holds true for all intervals of t. Therefore, the function 0 = 4e^(5t) - 2e^(2t) is indeed a solution to the given differential equation d²0/dt² + 50 = -7e^(2t).
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For the general rotation field F=axr, where a is a nonzero constant vector and r= (x,y,z), show that curl F=2a. Let a = = (a₁.a2,03) and write an explicit expression for F=axr. F=a₂z-a3y i+ -a₁z
The curl of the general rotation field F=axr, where a is a nonzero constant vector and r=(x,y,z), is equal to 2a.
This means that the curl of F, denoted as curl F, is a vector with components 2a₁, 2a₂, and 2a₃ in the x, y, and z directions, respectively.
To calculate the curl of F, we use the formula curl F = (∂F₃/∂y - ∂F₂/∂z)i + (∂F₁/∂z - ∂F₃/∂x)j + (∂F₂/∂x - ∂F₁/∂y)k. By substituting the components of F, which are F₁ = -a₃y, F₂ = a₂z, and F₃ = -a₁z, into the formula, we obtain (∂F₃/∂y - ∂F₂/∂z)i + (∂F₁/∂z - ∂F₃/∂x)j + (∂F₂/∂x - ∂F₁/∂y)k = (0 - a₂)i + (0 - 0)j + (0 - 0)k = -a₂i. Since the components of the curl are -a₂, 0, and 0, we can see that the curl of F is 2a.
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The curl of the general rotation field F=axr, where a is a nonzero constant vector and r=(x,y,z), is equal to 2a.
This means that the curl of F, denoted as curl F, is a vector with components 2a₁, 2a₂, and 2a₃ in the x, y, and z directions, respectively.
To calculate the curl of F, we use the formula curl F = (∂F₃/∂y - ∂F₂/∂z)i + (∂F₁/∂z - ∂F₃/∂x)j + (∂F₂/∂x - ∂F₁/∂y)k. By substituting the components of F, which are F₁ = -a₃y, F₂ = a₂z, and F₃ = -a₁z, into the formula, we obtain (∂F₃/∂y - ∂F₂/∂z)i + (∂F₁/∂z - ∂F₃/∂x)j + (∂F₂/∂x - ∂F₁/∂y)k = (0 - a₂)i + (0 - 0)j + (0 - 0)k = -a₂i. Since the components of the curl are -a₂, 0, and 0, we can see that the curl of F is 2a.
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Use the Laws of Logarithms to expand the expression.
a. Loga (x²/yz³)
b. Log √x√y√z
a. Loga (x²/yz³) = Loga x² - Loga yz³ [logarithm of quotient is equal to the difference of logarithm of numerator and logarithm of denominator]
Now, by the Laws of Logarithms, Loga (x²/yz³) can be written as: [tex]2Loga x - [3Loga y + Loga z³]b. Log √x√y√z = (1/2)Log x + (1/2)Log y + (1/2)Log z[/tex] [logarithm of product is equal to the sum of logarithm of factors]
Now, by the Laws of Logarithms, Log √x√y√z can be written as:[tex](1/2)Log x + (1/2)Log y + (1/2)Log z[/tex] [Note that square root of product of x, y and z is equal to product of square roots of x, y and z.]I hope this helps.
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1 Let r varies inversely as u, and r = 4 when u = 5. Find r if u = 1/6 1 If u =1/6, then r= _____₁ (Simplify your answer.)
K = r × u = 4 × 5 = 20.Now, u = 1/6, substitute this value in the above equation.r = k/u = 20/(1/6) = 120, if u = 1/6, then r = 120.
Given that r varies inversely as u and r = 4 when u = 5. To find the value of r when u = 1/6. Inversely proportional variables: When one variable increases and the other variable decreases, then two variables are said to be inversely proportional to each other. It can be shown as:r α 1/u ⇒ r = k/uwhere k is the constant of variation. Here, k = r × u. We know that when u = 5, r = 4. Therefore, k = r × u = 4 × 5 = 20.Now, u = 1/6, substitute this value in the above equation.r = k/u = 20/(1/6) = 120Hence, the value of r is 120 when u = 1/6.Answer:Therefore, if u = 1/6, then r = 120.
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Calculate the eigenvalues and the corresponding eigenvectors of the following matrix (a € R, bER\ {0}): a b A = ^-( :) b a
It appears to involve Laplace transforms and initial-value problems, but the equations and initial conditions are not properly formatted.
To solve initial-value problems using Laplace transforms, you typically need well-defined equations and initial conditions. Please provide the complete and properly formatted equations and initial conditions so that I can assist you further.
Inverting the Laplace transform: Using the table of Laplace transforms or partial fraction decomposition, we can find the inverse Laplace transform of Y(s) to obtain the solution y(t).
Please note that due to the complexity of the equation you provided, the solution process may differ. It is crucial to have the complete and accurately formatted equation and initial conditions to provide a precise solution.
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Find two linearly independent solutions of y′′+4xy=0y″+4xy=0 of the form
y1=1+a3x3+a6x6+⋯y1=1+a3x3+a6x6+⋯
y2=x+b4x4+b7x7+⋯y2=x+b4x4+b7x7+⋯
Enter the first few coefficients:
a3=a3=
a6=a6=
b4=b4=
b7=b7=
The two linearly independent solutions of the given differential equation are:
[tex]y1 = 1 - (2/3)x^3 + (4/45)x^6 + ...[/tex]
y2 = x
We have,
To find the coefficients for the linearly independent solutions of the given differential equation, we can use the power series method.
We start by assuming the solutions can be expressed as power series:
[tex]y1 = 1 + a3x^3 + a6x^6 + ...\\y2 = x + b4x^4 + b7x^7 + ...[/tex]
Now, we differentiate these series twice to find the corresponding derivatives:
[tex]y1' = 3a3x^2 + 6a6x^5 + ...\\y1'' = 6a3x + 30a6x^4 + ...[/tex]
[tex]y2' = 1 + 4b4x^3 + 7b7x^6 + ...\\y2'' = 12b4x^2 + 42b7x^5 + ...[/tex]
Substituting these expressions into the differential equation, we have:
[tex](y1'') + 4x(y1) = (6a3x + 30a6x^4 + ...) + 4x(1 + a3x^3 + a6x^6 + ...) = 0[/tex]
Collecting like terms, we get:
[tex]6a3x + 30a6x^4 + 4x + 4a3x^4 + 4a6x^7 + ... = 0[/tex]
To satisfy this equation for all values of x, each term must be individually zero.
Equating coefficients of like powers of x, we can solve for the coefficients:
For terms with x:
6a3 + 4 = 0
a3 = -2/3
For terms with [tex]x^4[/tex]:
30a6 + 4a3 = 0
30a6 - 8/3 = 0
a6 = 8/90 = 4/45
Similarly, we can find the coefficients for y2:
For terms with x³:
4b4 = 0
b4 = 0
For terms with [tex]x^6[/tex]:
4b7 = 0
b7 = 0
Therefore,
The coefficients are:
a3 = -2/3
a6 = 4/45
b4 = 0
b7 = 0
Thus,
The two linearly independent solutions of the given differential equation are:
[tex]y1 = 1 - (2/3)x^3 + (4/45)x^6 + ...[/tex]
y2 = x
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When the equation of the line is in the form y=mx+b, what is the value of **b**?
The regression equation is y = 1.1x - 0.7 and, the value of b is -0.7
How to determine the regression equatin and find bFrom the question, we have the following parameters that can be used in our computation:
(1, 0), (2, 3), (3, 1), (4, 4) and (5, 5)
Next, we enter the values in a graping tool where we have the following summary:
Sum of X = 15Sum of Y = 13Mean X = 3Mean Y = 2.6Sum of squares (SSX) = 10Sum of products (SP) = 11The regression equation is represented as
y = mx + b
Where
m = SP/SSX = 11/10 = 1.1
b = MY - bMX = 2.6 - (1.1*3) = -0.7
So, we have
y = 1.1x - 0.7
Hence, the value of b is -0.7
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8. The present value of an annuity is given. Find the periodic payment. (Round your final answer to two decimal places.)
Present value = $11,000, and the interest rate is 7.8% compounded monthly for 6 years.
9. Find the present value of the annuity that will pay $2000 every 6 months for 9 years from an account paying interest at a rate of 4% compounded semiannually. (Round your final answer to two decimal places.)
The answer are:
8.The periodic payment is approximately $861.88.
9.The present value of the annuity is approximately $1012.8.
What is the formula for the present value of an annuity?
The formula for the present value (PV) of an annuity is given by:
[tex]PV =\frac{ P(1 - (1 + r)^{-n}}{r}[/tex]
Where:
PV = Present Value
P = Periodic payment
r = Interest rate per period
n = Number of periods
8.In this case, we are given:
Present Value (PV) = $11,000
Interest Rate (r) = 7.8% = 0.078 (converted to decimal)
Number of Periods (n) = 6 years * 12 months/year = 72 months
Let's substitute the given values into the formula and solve for the periodic payment (P):
[tex]$11,000 =\frac{ P(1 - (1 + 0.078)^{-72})}{0.078}[/tex]
Now we can solve this equation to find the periodic payment:
[tex]{$11,000}*{0.078} = P(1 - (1 + 0.078)^{-72})[/tex]
[tex]858 = P(1 - 0.004481)\\P = \frac{858}{1 - 0.004481}\\P = \frac{858}{ 0.9955}\\ P= 861.88[/tex]
Therefore, the periodic payment is approximately $861.88.
9.To find the present value of an annuity, we can use the present value formula again.
In this case, we are given:
Periodic Payment (P) = $2000
Interest Rate (r) = 4% = 0.04 (converted to decimal)
Number of Periods (n) = 9 years * 2 semesters/year = 18 semesters
Let's substitute the given values into the formula and solve for the present value (PV):
[tex]PV =2000 *\frac{1 - (1 + 0.04)^{-18}}{0.04}[/tex]
Now we can solve this equation to find the present value (PV):
[tex]PV = $2000 *(1 - 1.04^{-18})\\ PV = $2000 * (1 - 0.4936)\\PV=$2000 * 0.5064\\ PV =$1012.8[/tex]
Therefore, the present value of the annuity is approximately $1012.8.
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ou wish to test the following claim (Ha) at a significance level of a 0.01 HPL - P2 HP> P2 The 1st population's sample has 126 successes and a sample size - 629, The 2nd population's sample has 60 successes and a sample size - 404 What is the test statistic (z-score) for this sample? (Round to 3 decimal places.
To obtain the test statistic (z-score) for this sample, use the formula:[tex]$$z=\frac{\hat{p_1}-\hat{p_2}}{\sqrt{\hat{p}(1-\hat{p})(\frac{1}{n_1}+\frac{1}{n_2})}}$$[/tex] where [tex]$\hat{p}$[/tex] is the pooled sample proportion,[tex]$n_1$[/tex] and $n_2$ [tex]$n_1$[/tex] are the sample sizes, [tex]$\hat{p_1}$ and $\hat{p_2}$[/tex] are the sample proportions of the two samples respectively.
[tex]$\hat{p}$[/tex] is calculated as:[tex]$$\hat{p}=\frac{x_1+x_2}{n_1+n_2}$$[/tex] where [tex]$x_1$ and $x_2$[/tex] are the number of successes in the first and second samples, respectively. Plugging in the given values, we get:[tex]$$\hat{p_1}=\frac{x_1}{n_1}=\frac{126}{629}[/tex] \approx [tex]0.200317$$$$\hat{p_2}=\frac{x_2}{n_2}=[/tex]\[tex]frac{60}{404}[/tex]\approx [tex]0.148515$$$$\hat{p}=\frac{x_1+x_2}{n_1+n_2}[/tex]=[tex]\frac{126+60}{629+404} \approx 0.1818$$[/tex] Substituting these values in the formula for $z$, we get:[tex]$$z=\frac{\hat{p_1}-\hat{p_2}}[/tex][tex](\frac{1}{n_1}+\frac{1}{n_2})}}$$$$[/tex] [tex]{\sqrt{\hat{p}(1-\hat{p})[/tex]=[tex]\frac{0.200317-0.148515}[/tex]{[tex]\sqrt{0.1818(1-0.1818)(\frac{1}{629}+\frac{1}{404})}}$$$$[/tex]\approx[tex]3.289$[/tex]
Rounding to three decimal places, the test statistic (z-score) for this sample is approximately equal to 3.289. Therefore, the correct answer is 3.289.
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Let D be the region bounded by a curve 2³+y³: = 3xy in the first quadrant. Find the area. of D (Hint: parametrise the curve so that y/x = t.)
Let us begin by sketching the curve of 2³ + y³ = 3xy in the first quadrant. Using the hint, we set y/x = t.
Now, y = tx.Substituting y = tx into the equation of the curve, we get:2³ + (tx)³ = 3x(tx)2³ + t³x³ = 3t²x³x³(3t² - 1) = 8We get x³ = 8 / (3t² - 1)Also, when x = 0, y = 0, and when y = 0, x = 0.
Hence, the region D can be expressed as the set:{(x,y): 0 ≤ x ≤ x_0, 0 ≤ y ≤ tx}where x_0 is a positive real number to be determined.
By definition, the area of D is given by ∬D dxdy, which can be expressed in terms of x_0 as:Area of D = ∫₀ˣ₀ ∫₀ᵗₓ₀ 1 dy dx
Let y = tx, then y/x = t and we have:y³ = t³x³Therefore:2³ + t³x³ = 3t²x³ ⇒ x³(3t² - 1) = 8 ⇒ x³ = 8 / (3t² - 1)Let f(t) = xₒ.
Then D is the region:{(x, y): 0 ≤ x ≤ xₒ, 0 ≤ y ≤ tx}Thus the area of D is given by:∬D dxdy = ∫₀ˣ₀ ∫₀ᵗₓ₀ 1 dy dx
Summary:Let y = tx, then y/x = t and we have:y³ = t³x³
Therefore:2³ + t³x³ = 3t²x³ ⇒ x³(3t² - 1) = 8 ⇒ x³ = 8 / (3t² - 1)Let f(t) = xₒ. Then D is the region:{(x, y): 0 ≤ x ≤ xₒ, 0 ≤ y ≤ tx}Thus the area of D is given by:∬D dxdy = ∫₀ˣ₀ ∫₀ᵗₓ₀ 1 dy dx
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Let h(x) = x² - 3 with po = 1 and p₁ = 2. Find på. (a) Use the secant method. (b) Use the method of False Position.
Using the secant method p_a is 1.75 and using the method of false position p_a is 1.75.
Given, h(x) = x^2 - 3 with p_0 = 1 and p_1 = 2.
We need to find p_a.
(a) Using the secant method
The formula for secant method is given by,
p_{n+1} = p_n - \frac{f(p_n) (p_n - p_{n-1})}{f(p_n) - f(p_{n-1})}
where n = 0, 1, 2, ...
Using the above formula, we get,
p_2 = p_1 - \frac{f(p_1) (p_1 - p_0)}{f(p_1) - f(p_0)}
\Rightarrow p_2 = 2 - \frac{(2^2 - 3) (2-1)}{(2^2-3) - ((1^2-3))}
\Rightarrow p_2 = 1.75
Therefore, p_a = 1.75.
(b) Using the method of false position
The formula for the method of false position is given by,
p_{n+1} = p_n - \frac{f(p_n) (p_n - p_{n-1})}{f(p_n) - f(p_{n-1})}
where n = 0, 1, 2, ...
Using the above formula, we get,
p_2 = p_1 - \frac{f(p_1) (p_1 - p_0)}{f(p_1) - f(p_0)}
\Rightarrow p_2 = 2 - \frac{(2^2 - 3) (2-1)}{(2^2-3) - ((1^2-3))}
\Rightarrow p_2 = 1.75
Therefore, p_a = 1.75.
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for a two-tailed hypothesis test for the pearson correlation, the null hypothesis states that
The specific null and alternative hypotheses for a hypothesis test will depend on the research question being investigated and the type of data being analyzed.
We have,
Equivalent expressions can be stated as the expressions which perform the same function despite their appearance. If two algebraic expressions are equivalent, they have the same value when we use the same variable value.
For a two-tailed hypothesis test, we know that, an appropriate null hypothesis indicating that the population correlation is equal to zero would be:
H₀: ρ = 0
where ρ represents the population correlation coefficient.
This null hypothesis states that there is no significant correlation between the two variables being analyzed.
In a two-tailed hypothesis test, the alternative hypothesis would be that there is a significant correlation, either positive or negative, between the two variables:
Hₐ: ρ ≠ 0
This alternative hypothesis states that there is a significant correlation between the two variables, but does not specify the direction of the correlation.
It's important to note that the specific null and alternative hypotheses for a hypothesis test will depend on the research question being investigated and the type of data being analyzed.
Additionally, the choice of null and alternative hypotheses will affect the statistical power of the test, which is the probability of correctly rejecting the null hypothesis when it is false.
Hence, the specific null and alternative hypotheses for a hypothesis test will depend on the research question being investigated and the type of data being analyzed.
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Complete Question:
For a two-tailed hypothesis test, which of the following would be an appropriate null hypothesis indicating that the population correlation is equal to o?
A. H₀: 1 = 2, B. H₀ : M₁ = M₂ C. H₀: O = 0
D. None of the options above are correct.
Find the Fourier series expansion of the function f(x) with period p = 21
1. f(x) = -1 (-2
2. f(x)=0 (-2
3. f(x)=x² (-1
4. f(x)= x³/2
5. f(x)=sin x
6. f(x) = cos #x
7. f(x) = |x| (-1
8. f(x) = (1 [1 + xif-1
9. f(x) = 1x² (-1
10. f(x)=0 (-2
f(x) = -1, f(x) = 0,No Fourier series expansion, No Fourier series expansion f(x) = (4/π) * (sin(x) - (1/3) * sin(3x) + (1/5) * sin(5x) - ...)f(x) = (a₀/2) + Σ(an * cos(n#x) + bn * sin(n#x))
Fourier series expansion represents a periodic function as a sum of sine and cosine functions. Let's find the Fourier series expansions for the given functions:
For the function f(x) = -1, the Fourier series expansion will have only a constant term. The expansion is f(x) = -1.
For the function f(x) = 0, which is a constant function, the Fourier series expansion will also have only a constant term. The expansion is f(x) = 0.
For the function f(x) = x², the Fourier series expansion can be found by calculating the coefficients. However, since the function is not periodic with a period of 21, it does not have a Fourier series expansion.
For the function f(x) = x³/2, similar to the previous function, it is not periodic with a period of 21, so it does not have a Fourier series expansion.
For the function f(x) = sin(x), which is periodic with a period of 2π, we can express it as a Fourier series expansion with coefficients of sin(nx) and cos(nx). In this case, the expansion is f(x) = (4/π) * (sin(x) - (1/3) * sin(3x) + (1/5) * sin(5x) - ...).
For the function f(x) = cos(#x), where "#" represents a constant, the Fourier series expansion will also have coefficients of sin(nx) and cos(nx). The expansion is f(x) = (a₀/2) + Σ(an * cos(n#x) + bn * sin(n#x)), where a₀ is the average value of f(x) over a period and an, bn are the Fourier coefficients.
For the function f(x) = |x|, which is an absolute value function, the Fourier series expansion can be calculated piecewise for different intervals. However, since the function is not periodic with a period of 21, it does not have a simple Fourier series expansion.
For the function f(x) = (1 + x)^(if-1), the Fourier series expansion depends on the specific value of "if." Without knowing the value, it is not possible to determine the exact Fourier series expansion.
For the function f(x) = 1/x², similar to the previous cases, it is not periodic with a period of 21, so it does not have a Fourier series expansion.
For the function f(x) = 0, which is a constant function, the Fourier series expansion will have only a constant term. The expansion is f(x) = 0.
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Let f(x)=(x+2)(x+6)5
F(x)=
Use the chain rule to find the derivative of f'(x) = 4 (-6x3-9x9)19, You do not need to expand out your answer.
F’(x)=
To find the derivative of the function [tex]f(x) = (x+2)(x+6)^5,[/tex] we can use the chain rule. By differentiating the outer function and then multiplying it by the derivative of the inner function, we can determine the derivative of f(x). In this case, the derivative is f'(x) = [tex]4(-6x^3 - 9x^9)^19.[/tex]
Let's find the derivative of the function f(x) = (x+2)(x+6)^5 using the chain rule.
The outer function is (x+2) and the inner function is (x+6)^5.
Differentiating the outer function with respect to its argument, we get 1.
Now, we need to multiply this by the derivative of the inner function.
Differentiating the inner function, we get d/dx((x+6)^5) = 5(x+6)^4.
Multiplying the derivative of the outer function by the derivative of the inner function, we have:
[tex]f'(x) = 1 * 5(x+6)^4 = 5(x+6)^4.[/tex]
Finally, we can simplify the expression:[tex]f(x) = (x+2)(x+6)^5[/tex]
[tex]f'(x) = 5(x+6)^4.[/tex]
Therefore, the derivative of the function f(x) =[tex](x+2)(x+6)^5 is f'(x)[/tex]= [tex]5(x+6)^4.[/tex]
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Determine the point of intersection of the lines r(t) = (4 +1,-- 8 + 91.7) and (u) = (8 + 4u. Bu, 8 + U) Answer 2 Points Ке Keyboard St
Therefore, the point of intersection of the lines r(t) and u(t) is (24, 172, 12).
To determine the point of intersection of the lines r(t) = (4 + t, -8 + 9t) and u(t) = (8 + 4u, Bu, 8 + u), we need to find the values of t and u where the x, y, and z coordinates of the two lines are equal.
The x-coordinate equality gives us:
4 + t = 8 + 4u
t = 4u + 4
The y-coordinate equality gives us:
-8 + 9t = Bu
9t = Bu + 8
The z-coordinate equality gives us:
-8 + 9t = 8 + u
9t = u + 16
From the first and second equations, we can equate t in terms of u:
4u + 4 = Bu + 8
4u - Bu = 4
From the second and third equations, we can equate t in terms of u:
Bu + 8 = u + 16
Bu - u = 8
Now we have a system of two equations with two unknowns (u and B). Solving these equations will give us the values of u and B. Multiplying the second equation by 4 and adding it to the first equation to eliminate the variable B, we get:
4u - Bu + 4(Bu - u) = 4 + 4(8)
4u - Bu + 4Bu - 4u = 4 + 32
3Bu = 36
Bu = 12
Substituting Bu = 12 into the second equation, we have:
12 - u = 8
-u = 8 - 12
-u = -4
u = 4
Substituting u = 4 into the first equation, we have:
4(4) - B(4) = 4
16 - 4B = 4
-4B = 4 - 16
-4B = -12
B = 3
Now we have the values of u = 4 and B = 3. We can substitute these values back into the equations for t:
t = 4u + 4
t = 4(4) + 4
t = 16 + 4
t = 20
So the values of t and u are t = 20 and u = 4, respectively.
Now we can substitute these values back into the original equations for r(t) and u(t) to find the point of intersection:
r(20) = (4 + 20, -8 + 9(20))
r(20) = (24, 172)
u(4) = (8 + 4(4), 3(4), 8 + 4)
u(4) = (24, 12, 12)
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8. On average 1,500 pupils join PMU each year for registration and pay SR4.00 for drinking-water on campus. The number of pupils q willing to join PMU at drinking- water price p is q(p) = 600(5- Vp). Is the demand elastic, inelastic, or unitary at p=4?
A 1% increase in price will result in a less than 1% decrease in quantity demanded, and vice versa.
To determine the elasticity of demand at a price of p=4, we need to calculate the price elasticity of demand using the formula:
Price elasticity of demand = (% change in quantity demanded / % change in price)
Since we are given a specific price of p=4, we need to calculate the corresponding quantity demanded using the demand function:
q(4) = 600(5 - sqrt(4)) = 600(3) = 1800
Now, let's imagine that the price of drinking-water on campus increases from p=4 to p=5. The new quantity demanded would be:
q(5) = 600(5 - sqrt(5)) = 600(2.76) = 1656
Using these values, we can calculate the price elasticity of demand:
Price elasticity of demand = ((1656-1800)/((1656+1800)/2)) / ((5-4)/((5+4)/2)) = -0.95
Since the price elasticity of demand is less than 1 in absolute value, we can conclude that the demand for drinking-water on campus at PMU is inelastic at a price of p=4. This means that a 1% increase in price will result in a less than 1% decrease in quantity demanded, and vice versa.
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to V 14. In each of the following, prove that the given lines are mutually perpendicular: -1 3x + y - 5z + 1 = 0, a) = ² = and
To prove that the lines -1 + 3x + y - 5z + 1 = 0 and a) = ² = are mutually perpendicular, we will show that their direction vectors are orthogonal.
To determine if two lines are mutually perpendicular, we need to examine the dot product of their direction vectors. The given lines can be rewritten in the form of directional vectors:
Line 1 has a direction vector [3, 1, -5], and Line 2 has a direction vector [a, b, c].
To check if these vectors are perpendicular, we calculate their dot product: (3)(a) + (1)(b) + (-5)(c). If this dot product equals zero, the lines are mutually perpendicular.
Therefore, the condition for perpendicularity is 3a + b - 5c = 0. If this equation holds true, then the lines -1 + 3x + y - 5z + 1 = 0 and a) = ² = are mutually perpendicular.
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Assume you are using a significance level of a 0.05) to test the claim that < 13 and that your sample is a random sample of 41 values. Find the probability of making a type II error (failing to reject a false null hypothesis), given that the population actually has a normal distribution with μ-8 and 7J B = |
The probability of making a type II error, failing to reject a false null hypothesis, is influenced by the specific alternative hypothesis being tested. In this case, when testing the claim that the population mean is less than 13, given a random sample of 41 values from a normally distributed population with a mean of μ = 8 and standard deviation σ = 7, the probability of a type II error can be calculated.
To calculate the probability of a type II error, we need to determine the specific alternative hypothesis and the corresponding critical value. Since we are testing the claim that the population mean is less than 13, the alternative hypothesis can be expressed as H₁: μ < 13.
Next, we need to find the critical value corresponding to the significance level (α) of 0.05. Since this is a one-tailed test with the alternative hypothesis indicating a left-tailed distribution, we can find the critical value using a z-table or calculator. With a significance level of 0.05, the critical z-value is approximately -1.645.
Using the given values, we can calculate the z-score for the critical value of -1.645 and find the corresponding cumulative probability from the z-table or calculator. This probability represents the probability of observing a value less than 13 when the population mean is actually 8.
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Given P(A) = 0.508, find the probability of the complementary event. O 0.332 O None of these O 0.492 O 0.376 O 0.004
The probability of the complementary event is 0.492. Option a is correct.
The probability of the complementary event, denoted as P(A'), is equal to 1 minus the probability of event A.
P(A') = 1 - P(A)
In this case, we are given that P(A) = 0.508. To find the probability of the complementary event, we subtract the probability of event A from 1. Therefore, we can calculate the probability of the complementary event as:
P(A') = 1 - 0.508 = 0.492
Therefore, the probability of the complementary event is calculated as 1 - 0.508 = 0.492.
Hence, the correct answer is A. 0.492.
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a) Evaluate the integral of the following tabular data х 0 0.15 0.32 0.48 0.64 0.7 0.81 0.92 1.03 3.61
f(x) 3.2 11.9048 13.7408 15.57 19.34 21.6065 23.4966 27.3867 31.3012 44.356 using a combination of the trapezoidal and Simpson's rules. b) How to get a higher accuracy in the solution? Please explain in brief. c) Which method provides more accurate result trapezoidal or Simpson's rule? d) How can you increase the accuracy of the trapezoidal rule? Please explain your comments with this given data.
The value of the integral of the tabular data using the combination of the trapezoidal and Simpson's rule is 56.1874.
How to find?The interval limits and values of $f(x)$ are listed in the table below.
Adding up the individual integrals calculated using both the trapezoidal and Simpson's rule we get:
$\begin{aligned} &\int_{0}^{3.61} f(x) dx\\
=&T_1 + T_2 + T_3 + T_4 + S_1 + S_2\\
=&2.432 + 3.2768 + 3.9435 + 36.3571 + 2.4469 + 3.2451 + 3.8845 + 3.6015\\
=&56.1874 \end{aligned}$.
Therefore, the value of the integral of the tabular data using the combination of the trapezoidal and Simpson's rule is 56.1874.
b) How to get a higher accuracy in the solution?One way to increase the accuracy of the solution is to use more intervals.This will help capture the behavior of the function in more detail, resulting in a more accurate approximation of the integral. Another way to increase accuracy is to use a higher-order method, such as Simpson's 3/8 rule or Gaussian quadrature.c) Which method provides a more accurate result: trapezoidal or Simpson's rule?Simpson's rule provides a more accurate result than the trapezoidal rule, because it uses a higher-order polynomial approximation of the function within each interval. Specifically, Simpson's rule uses a quadratic polynomial, while the trapezoidal rule uses a linear polynomial.d) How can you increase the accuracy of the trapezoidal rule?To increase the accuracy of the trapezoidal rule, you can use more intervals. This will help capture the behavior of the function in more detail, resulting in a more accurate approximation of the integral. Alternatively, you can use a higher-order method, such as Simpson's 3/8 rule or Gaussian quadrature.To know more on Trapezoidal rule visit:
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Using Green's function, evaluate f xdx + xydy, where e is the triangular curve consisting of the line segments from (0,0) to (1,0), from (1,0) to (0,1) and from (0,1) to (0.0).
To evaluate the integral ∫∫ f(x) dx + f(y) dy over the triangular curve e, we can use Green's theorem.
Green's theorem relates the line integral of a vector field over a closed curve to the double integral of the curl of the vector field over the region enclosed by the curve. Let's denote the vector field as F(x, y) = (f(x), f(y)). The curl of F is given by ∇ x F, where ∇ is the del operator. In two dimensions, the curl is simply the z-component of the cross product of the del operator and the vector field, which is ∇ x F = (∂f(y)/∂x - ∂f(x)/∂y).
Applying Green's theorem, the double integral ∫∫ (∂f(y)/∂x - ∂f(x)/∂y) dA over the region enclosed by the triangular curve e is equal to the line integral ∫ f(x) dx + f(y) dy over the curve e. Since the triangular curve e is a simple closed curve, we can evaluate the double integral by parameterizing the region and computing the integral. First, we can parametrize the triangular region by using the standard parametrizations of each line segment. Let's denote the parameters as u and v. The parameterization for the triangular region can be written as:
x(u, v) = u(1 - v)
y(u, v) = v
The Jacobian of this transformation is |J(u, v)| = 1.
Next, we substitute these parametric equations into the expression for ∂f(y)/∂x - ∂f(x)/∂y and evaluate the double integral:
∫∫ (∂f(y)/∂x - ∂f(x)/∂y) dA
= ∫∫ (f'(y) - f'(x)) |J(u, v)| du dv
= ∫∫ (f'(v) - f'(u(1 - v))) du dv
To compute this integral, we need to know the function f(x) or f(y) and its derivative. Without that information, we cannot provide the exact numerical value of the integral. However, you can substitute your specific function f(x) or f(y) into the above expression and evaluate the integral accordingly.
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Compute the following determinants using the permutation expansion method. (Your can check your answers by also computing them via the Gaussian elimination method.) -8 7 5 0 0-1 a) 2 -5 -6 b) -1 4 -2 9 4 2 3 3
Using the permutation expansion method, we get the main answer as follows:
Simplifying the above equation, we get:$\det(B) = -19 - 52 - 6 + 16$$\det(B) = -61$Therefore, the main answer is -61.
Summary: The value of the determinant of the matrix A is 31 and the value of the determinant of the matrix B is -61.
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The cylinder below has a radius of 4cm and the length of 11cm
The volume of the cylinder is equal to 553 cm³.
How to calculate the volume of a cylinder?In Mathematics and Geometry, the volume of a cylinder can be calculated by using this formula:
Volume of a cylinder, V = πr²h
Where:
V represents the volume of a cylinder.h represents the height or length of a cylinder.r represents the radius of a cylinder.By substituting the given side lengths into the volume of a cylinder formula, we have the following;
Volume of cylinder, V = 3.14 × 4² × 11
Volume of cylinder, V = π × 16 × 11
Volume of cylinder, V = 552.64 ≈ 553 cm³.
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Missing information:
The question is incomplete and the complete question is shown in the attached picture.
"Is there significant evidence at 0.05 significance level to conclude that population A has a larger mean than population B?" Translate it into the appropriate hypothesis. A. Ηο: μΑ ≥ μΒ B. Ηο: μΑ > μΒ C. Ha: μΑ > μΒ D. Ha: μΑ ≠ μΒ
The appropriate hypothesis can be translated as follows: C. Ha: μΑ > μΒ.Explanation:
We can interpret this problem using the hypothesis testing framework. We can start by defining the null hypothesis and the alternative hypothesis. Then we can perform a hypothesis test to see if there is enough evidence to reject the null hypothesis and accept the alternative hypothesis.H0: μA ≤ μBHA: μA > μBWe are testing if population A has a larger mean than population B.
The alternative hypothesis should reflect this. The null hypothesis states that there is no difference between the means or that population A has a smaller or equal mean than population B. The alternative hypothesis states that population A has a larger mean than population B. The appropriate hypothesis can be translated as follows:Ha: μA > μBWe can then use a t-test to test the hypothesis.
If the p-value is less than the significance level (0.05), we can reject the null hypothesis and conclude that there is significant evidence that population A has a larger mean than population B. If the p-value is greater than the significance level (0.05), we fail to reject the null hypothesis and do not have enough evidence to conclude that population A has a larger mean than population B.
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If [u, v, w] = 11, what is [w-v, u, w]? Select one: a.There is not enough information to say. b.22 c. 11 d.-22 e.0 Clear my choice
Given: [u, v, w] = 11To find: [w-v, u, w]Solution:In the expression [w-v, u, w], we have to replace the values of w, v and u.
Substituting w = 11, u = v = 0 in the given expression, we get;[w-v, u, w] = [11 - 0, 0, 11] = [11, 0, 11]Therefore, the answer is [11, 0, 11].Hence, the correct option is not (a) and the answer is [11, 0, 11].11 are provided for [u, v, and w].Find [w-v, u, w]The values of w, v, and u in the expression [w-v, u, w] must be modified.By replacing w, u, and v with 11, 0, and 0, respectively, in the previous formula, we arrive at [w-v, u, w] = [11 - 0, 0, 11] = [11, 0, 11].Therefore, the answer is [11, 0, 11].As a result, option (a) is erroneous and the answer of [11, 0, 11] is the right one.
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The answer for the given matrix is [11, 0, 11]. As a result, option (a) is erroneous and the answer of [11, 0, 11] is the right one.
Given: [u, v, w] = 11
To find: [w-v, u, w]
In the expression [w-v, u, w], we have to replace the values of w, v and u.
Substituting ,
w = 11,
u = v = 0 in the given expression, we get;
[w-v, u, w]
= [11 - 0, 0, 11]
= [11, 0, 11]
Therefore, the answer is [11, 0, 11].
Hence, the correct option is not (a) and the answer is [11, 0, 11]. 11 are provided for [u, v, and w].
Find [w-v, u, w]
The values of w, v, and u in the expression [w-v, u, w] must be modified. By replacing w, u, and v with 11, 0, and 0, respectively, in the previous formula, we arrive at [w-v, u, w] = [11 - 0, 0, 11] = [11, 0, 11].
Therefore, the answer is [11, 0, 11].As a result, option (a) is erroneous and the answer of [11, 0, 11] is the right one.
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Question 7 (3 points) What is the purpose of the discriminant? Provide a diagram and example with your explanation.
The value of the discriminant is positive, there are two distinct real roots.
The discriminant is an expression that appears under the radical sign in the quadratic formula. It helps determine the nature of roots of a quadratic equation.
When the value of the discriminant is positive, it indicates that the quadratic equation has two distinct real roots.
When the value of the discriminant is zero, it indicates that the quadratic equation has one repeated real root.
When the value of the discriminant is negative, it indicates that the quadratic equation has two complex roots that are not real numbers.
The diagram below is a visual representation of the nature of the roots of a quadratic equation based on the value of the discriminant.
[tex]\Delta[/tex] = b2 - 4acFor instance, consider the quadratic equation below: x2 + 5x + 6 = 0.
The value of the discriminant is:b2 - 4ac= 52 - 4(1)(6)= 25 - 24= 1
Since the value of the discriminant is positive, there are two distinct real roots.
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find the fourier series of the function f on the given interval. f(x) = 0, −π < x < 0 1, 0 ≤ x < π
The Fourier series of the function f(x) on the interval -π < x < π is f(x) = (1/π) + ∑[(2/π) [1 - cos(nπ)] sin(nx)].
What is the Fourier series of the function f(x) = 0, −π < x < 0; 1, 0 ≤ x < π on the given interval?To find the Fourier series of the function f(x) on the given interval, we can use the formula for the Fourier coefficients.
Since f(x) is a piecewise function with different definitions on different intervals, we need to determine the coefficients for each interval separately.
For the interval -π < x < 0, f(x) is equal to 0. Therefore, all the Fourier coefficients for this interval will be 0.
For the interval 0 ≤ x < π, f(x) is equal to 1. To find the coefficients for this interval, we can use the formula:
a₀ = (1/π) ∫[0,π] f(x) dx = (1/π) ∫[0,π] 1 dx = 1/π
aₙ = (1/π) ∫[0,π] f(x) cos(nx) dx = (1/π) ∫[0,π] 1 cos(nx) dx = 0
bₙ = (1/π) ∫[0,π] f(x) sin(nx) dx = (1/π) ∫[0,π] 1 sin(nx) dx = (2/π) [1 - cos(nπ)]
Therefore, the Fourier series of f(x) on the given interval is:
f(x) = (1/π) + ∑[(2/π) [1 - cos(nπ)] sin(nx)]
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A researcher knows that the weights of 6 year olds are normally distributed with \mu = 20.9 and \sigma = 3.2. It is claimed that all 6 year old children weighing less than 18.2 kg can be considered underweight and therefore undernourished. If a sample of n = 9 children is therefore selected from this population, find the probability that their average weight is less tha or equal to 18.2kg?
The probability that the average weight of a sample of 9 six-year-old children is less than or equal to 18.2 kg, given a population with a mean of 20.9 kg and a standard deviation of 3.2 kg, can be determined using the sampling distribution of the sample mean.
In this scenario, we are dealing with the distribution of sample means, which follows the Central Limit Theorem. The Central Limit Theorem states that when the sample size is sufficiently large, the sampling distribution of the sample mean will be approximately normally distributed, regardless of the shape of the population distribution.
To find the probability that the average weight of a sample of 9 children is less than or equal to 18.2 kg, we need to calculate the z-score for this value. The z-score measures the number of standard deviations a value is from the mean. Using the formula z = (x - μ) / (σ / sqrt(n)), where x is the sample mean, μ is the population mean, σ is the population standard deviation, and n is the sample size, we can calculate the z-score.
For this problem, x is 18.2 kg, μ is 20.9 kg, σ is 3.2 kg, and n is 9. Substituting these values into the formula, we find that the z-score is z = (18.2 - 20.9) / (3.2 / sqrt(9)) = -2.7 / 1.066 = -2.53 (rounded to two decimal places).
Next, we can use a standard normal distribution table or a statistical software to find the probability associated with a z-score of -2.53. The probability corresponds to the area under the standard normal curve to the left of -2.53. By looking up this value, we find that the probability is approximately 0.0058.
Therefore, the probability that the average weight of a sample of 9 six-year-old children is less than or equal to 18.2 kg is approximately 0.0058, or 0.58%.
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"
Need help solving problem
D Question 17 Solve the equation. (64) x+1= X-1 - 27 O {-1)
Thus, the solution to the equation is: [tex]x = -92/63.[/tex]
To solve the equation [tex](64)x+1 = x-1 - 27[/tex], we can follow these steps:
Simplify both sides of the equation:
[tex]64(x+1) = x-1 - 27[/tex]
Distribute 64:
[tex]64x + 64 = x - 1 - 27[/tex]
Combine like terms:
[tex]64x + 64 = x - 28[/tex]
Subtract x from both sides and subtract 64 from both sides to isolate the variable:
[tex]64x - x = -28 - 64[/tex]
[tex]63x = -92[/tex]
Divide both sides by 63 to solve for x:
[tex]x = -92/63[/tex]
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