The equation of the line that passes through the point (-4 , 3) with slope ₋2 is:
y = ₋2x ₋ 5
Given the points are (₋4 , ₋3) and (4 , 1)
The gradient of the line through the points' locations (x₁ , y₁) and (x₂ , y₂) is: y₂ ₋ y₁/x₂ ₋ x₁
Substitute the given points in the above formula:
=1 ₋(₋3)/4 ₋ (₋4)
=4/8
=1/2
If the product of the slopes of two perpendicular lines is -1, then the slope of the perpendicular line is ₋1/1/2 = ₋2
Therefore the line that passes through the point (-4 , 3) with slope ₋2 has equation:
y ₋ 3 = ₋2(x ₊ 4)
y = ₋2x ₋ 8 ₊ 3
y = ₋2x ₋ 5
Hence we get the expression as y = ₋2x ₋ 5
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Find the equivalent exponential expression. [(-4)^5]^3
Answer:
[tex]-4^{15}[/tex]
Step-by-step explanation:
Given expression:
[tex]\left[(-4)^5\right]^3[/tex]
[tex]\textsf{Apply exponent rule} \quad (a^b)^c=a^{bc}:[/tex]
[tex]\implies (-4)^{(5 \times 3)}[/tex]
[tex]\implies (-4)^{15}[/tex]
[tex]\textsf{Apply exponent rule} \quad (-a)^n=-a^n,\:\: \textsf{ if }n \textsf{ is odd}:[/tex]
[tex]\implies -4^{15}[/tex]
Use separation of variables to solve the differential equation with initial condition y(1)=-2
Answer: A
Step-by-step explanation:
[tex]\frac{dy}{dx}=\frac{1+x}{xy} \\ \\ y \text{ } dy=\frac{1+x}{x} \text{ } dx \\ \\ y \text{ } dy= \left(\frac{1}{x}+1 \right) \text{ } dx \\ \\ \int y \text{ } dy=\int \left(\frac{1}{x}+1 \right) \text{ } dx \\ \\ \frac{y^{2}}{2}=\ln \left| x \right|+x+C[/tex]
Substituting in the initial condition,
[tex]\frac{(-2)^{2}}{2}=\ln \left|1 \right|+1+C\\\\2=1+C\\\\C=1[/tex]
So,
[tex]\boxed{\frac{1}{2}y^{2}=\ln \left|x \right|+x+1}[/tex]
Answer:
[tex]\textsf{A.} \quad \dfrac{1}{2}y^2=\ln|x|+x+1[/tex]
Step-by-step explanation:
To solve the differential equation:
Rearrange the given equation to get all the terms containing y on the left side, and all the terms containing x on the right side:
[tex]\dfrac{dy}{dx}=\dfrac{1+x}{xy}[/tex]
[tex]\implies y\dfrac{dy}{dx}=\dfrac{1+x}{x}[/tex]
[tex]\implies y\:dy=\dfrac{1+x}{x}\:dx[/tex]
[tex]\implies y\:dy=\dfrac{1}{x}+\dfrac{x}{x}\:dx[/tex]
[tex]\implies y\:dy=\left(\dfrac{1}{x}+1\right)\:dx[/tex]
Integrate both sides:
[tex]\displaystyle \implies \int y\:dy=\int \left(\dfrac{1}{x}+1\right)\:dx[/tex]
[tex]\implies \dfrac{1}{2}y^2=\ln|x|+x+C[/tex]
Find the value of C using the given values of x and y:
when x = 1, y = -2
[tex]\implies \dfrac{1}{2}(-2)^2=\ln|1|+1+C[/tex]
[tex]\implies 2=0+1+C[/tex]
[tex]\implies C=1[/tex]
Substitute the found value of C:
[tex]\implies \dfrac{1}{2}y^2=\ln|x|+x+1[/tex]
The point stope form of the equation of the line that passes through (-4,-3) and (12, 1) is y-1 = (x-12). What is
the standard form of the equation for this line?
Answer:
x-4y = 8
Step-by-step explanation:
The standard form for the equation of a line is Ax+By=C where A is a positive integer and B and C are integers
First we need to find the slope
m = ( y2-y1)/(x2-x1)
m = (-3-1) / (-4-12)
=-4/-16
= 1/4
The point slope form is
y-1 = 1/4(x-12)
Multiply each side by 4
4(y-1 )= 4*1/4(x-12)
4(y-1) = x-12
4y -4 = x-12
Subtract 4y from each side
4y-4-4y = x-4y -12
-4 = x-4y -12
Add 12 to each side
-4+12 = x-4y -12+12
8 = x-4y
x-4y = 8
The standard form is x-4y = 8
Given lJK , which is an isosceles right triangle, IJ=4 and m
Answer:
[tex]IK = IJ\sqrt{2}[/tex]
Step-by-step explanation:
First, it's an isosceles right triangle, which means there is a 90 degree angle and the two other angles are equal.
To find what the two other angles are, you can set up a quick equation:
let x be angle
90 + 2x = 180, since the sum of the interior angles in a triangle is always 180.
then,
[tex]2x = 90\\x = 45[/tex]
Therefore, the triangle is a 45-45-90 triangle.
You have the measure of IJ, which is one of the legs of the triangle. The way you can find the length of the hypotenuse of a 45-45-90 triangle is by multiplying the length of one of the legs by [tex]\sqrt{2}[/tex].
This means the equation is [tex]IK = IJ\sqrt{2}[/tex]
Answer:
[tex]\sf IK=IJ\sqrt{2}[/tex]
Step-by-step explanation:
The interior angles of a triangle sum to 180°. Therefore, if ΔIJK is an isosceles right triangle where m∠J = 90°:
vertex is m∠J = 90°two base angles are m∠K and m∠ITo calculate the base angles:
⇒ m∠K + m∠I + m∠J = 180°
⇒ m∠K + m∠I + 90° = 180°
⇒ m∠K + m∠I = 90°
⇒ m∠K = m∠I = 45°
Therefore, IJ and JK are the legs of the right triangle and IK is the hypotenuse.
To find the length of the hypotenuse, use Pythagoras' Theorem.
Pythagoras’ Theorem: [tex]\sf a^2+b^2=c^2[/tex]
(where a and b are the legs, and c is the hypotenuse, of a right triangle)
Given:
a = IJ = 4b = JK = 4c = IKSubstitute the given values into the formula and solve for IK:
[tex]\sf \implies IJ^2+JK^2=IK^2[/tex]
[tex]\sf \implies 4^2+4^2=IK^2[/tex]
[tex]\sf \implies IK^2=32[/tex]
[tex]\implies \sf IK=\sqrt{32}[/tex]
[tex]\implies \sf IK=\sqrt{16 \cdot 2}[/tex]
[tex]\implies \sf IK=\sqrt{16}\sqrt{2}[/tex]
[tex]\implies \sf IK=4\sqrt{2}[/tex]
As IJ = 4 then:
[tex]\implies \sf IK=IJ\sqrt{2}[/tex]
What are the slope and the y-intercept of the linear function that is represented by the graph?
answer C is “The slope is Negative three-fourths, and the y-intercept is 3.”
answer D is “The slope is Negative three-fourths, and the y-intercept is 4”
Answer:
“The slope is Negative three-fourths, and the y-intercept is 3.”
Step-by-step explanation:
the graph passes through the points (0 , 3) and (4 , 0)
Then
The slope is :
[tex]=\frac{0-3}{4-0}[/tex]
[tex]=-\frac{3}{4}[/tex]
Since , the graph passes through (0 , 3)
THen the y-intercept is 3.
If log_2(3) = a and log_2(10) = b then an expression for log_2(90) is
[tex]\log_290=\log_2(3^2\cdot10)=\log_23^2+\log_210=2\log_23+b=2a+b[/tex]
How many solutions are there for the system of equations of shown on the graph?
O No solution
O One solution
O Two solutions
O Infinitely many solutions
Answer:
one solution
(second option listed)
Step-by-step explanation:
We can that these two lines, each representing one equation/function, only meet at one specific value.
In a system of equations, we are essentially looking for a solution that works for both equations.
So, if both lines share a point/value (meaning they intersect), that point is a solution to the system of equations.
Because these lines only overlap at one point, this system of equations has one solution.
Answer:
1 solution (C)
Explanation:
They have one point of intersection.
Graph the system of inequalities presented here on your own paper; then use your graph to answer the following questions:
Y > -4x-1
y<3/2x-1
Part A:
Describe the graph of the system; including shading and the types of lines graphed Provide a description of the solution area: (6 points)
Part B: Is the point (-1,1) included in the solution area for the system? Justify your answer mathematically
It is to be noted that the Graph of the indicated inequalities is herewith attached.
What is the graph of the system?As indicated above, this is a graph of a system of inequalities, where:
Y > -4x-1 and
y <3/2x-1
Is the point (-1,1) included in the solution area for the system?No. the point (-1, 1) is not included in the solution area for the system.
Step 1 - Plug in (-1, 1) in to the first equation. where y = -1 and x = 1.
Hence,
-1 > -4(1) -1
4 +1 < -1
4 < 0 (This is False)
Step 2 - Plug in (-1, 1) in to the second equation. where y = -1 and x = 1.
-1 < (3/2)(1) - 1
-1 < 1/2 (This is true).
Thus (-1, 1) is only partially included in the solution area.)
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A boy earned $435, $230, and $562 for 3 consecutive months. How much did he earn in the fourth month so that he gets an average earning of $500 in these 4 months?
Answer:
The boy earned $773 in the fourth month.
Step-by-step explanation:
1) First, we need to find the total amount of money he needs so that the average of the 4 months would be 500. We can do this by multiplying 500 and 4 (essentially, we are working backward when trying to find the average) [tex]500*4=2000[/tex] So, we know that the boy needs $2000 total by the end of the fourth month to make the 4-month average $500.
2) Now, we know how much the boy made for the first 3 consecutive months, so add those numbers up. [tex]435+230+562=1227[/tex] So within the first 3 months, the boy made $1227.
3) Finally, we know how much money the boy needs by the end of the fourth month and how much he made by the third, so to figure out how much he earned in the fourth month, we can simply subtract $1227 from $2000. [tex]2000-1227=773[/tex]
Genesis is going to see a movie and is taking her 3 kids. Each movie ticket costs $15 and there are an assortment of snacks available to purchase for $3 each. How much total money would Genesis have to pay for her family if she were to buy 6 snacks for everybody to share? How much would Genesis have to pay if she bought x snacks for everybody to share?
Total cost with 6 snacks:
Total cost with x snacks:.
need answer quickly as possibly!
Answer #1:
The total cost with 6 snacks is $78.
Answer #2:
The total cost with x snacks is represented by the expression 60 + 3x.
Step-by-step explanation for answer #1:
Starting off with what we know:
Genesis has 3 kids, so there are 4 people total including Genesis.Each movie ticket is $15.Each snack is $3.To answer the question of how much money Genesis would have to pay for her family if she bought 6 snacks for everybody, first consider the total cost of the movie tickets.
Multiply the total number of people by the cost of each movie ticket:
[tex]4* 15=60[/tex]
Genesis will be paying $60 total only to see the movie with her 3 kids. To find the total cost of both seeing the movie and buying 6 snacks, simply multiply the number of snacks by the cost of each snack:
[tex]6 * 3=18[/tex]
Then, add the total cost of the movie tickets by the total cost of the snacks to achieve your first answer:
[tex]60+18=78[/tex]
The total cost with 6 snacks is $78.
Step-by-step explanation for answer #2:
To find the total cost of "x" snacks (x is being used as a variable term to represent a quantity subject to change), we can create an algebraic expression.
Let "t" represent the total cost.
Since we've established that finding the total cost is just a matter of adding the total cost of movie tickets (60) and the total cost of snacks (3 multiplied by the number of snacks), let "x" represent the number of snacks in the equation to find the total cost with x snacks:
[tex]60+3x=t[/tex]
The total cost with x snacks is represented by the expression 60 + 3x.
Based on the given data, the total cost of 6 snacks: $78. total cost with x snacks: $60 + (3x).
Use the concept of multiplication defined as:
To multiply means to add a number to itself a particular number of times. Multiplication can be viewed as a process of repeated addition.
Given that,
Genesis is going to see a movie with her 3 kids.
Each movie ticket costs $15.
There are snacks available for purchase at $3 each.
Genesis wants to buy 6 snacks for everybody to share.
To calculate the total cost for Genesis and her 3 kids,
Consider the movie tickets and the snacks.
The cost of the movie tickets can be calculated by multiplying the number of people (4 in this case) by the cost of each ticket ($15).
So, Genesis would have to pay 4 x $15 = $60 for the movie tickets.
Now, let's calculate the total cost if Genesis were to buy 6 snacks for everybody to share.
Each snack costs $3, so 6 snacks would cost 6 x $3 = $18.
Therefore, the total cost for Genesis would be,
$60 (movie tickets) + $18 (snacks) = $78.
If we consider x snacks for everybody to share,
The total cost for snacks would be x $3.
So, the final cost would be $60 (movie tickets) + (3x) (snacks).
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Divya is solving the following equation for w. 3w -√20-5w = 2 Her first few steps are given below. 3w = √20-5w+2 3w-2= √/20 - 5w (3w - 2)² =(√20 – 5w) 9w² - 12w + 4 = 20 - 5w Is it necessary for Divya to check her answers for extraneous solutions?
Divya should not check her answers for extraneous solution as it's not required in this case.
What is an extraneous solution?It should be noted that an extraneous solution simply means is the transformed equation that's isn't the root of the original equation.
Based on the information given, Divya had already solved the equation and it's not necessary to check the answers for extraneous solution.
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Answer:
No, she shouldn't.
A sequence has three terms.
Its term-to-term rule is
multiply by 6 and then add 13
a) The first term of the sequence is -2
Work out the third term.
b) The order of the three terms of the sequence is reversed.
Describe the term-to-term rule of the new sequence.
Answer:
a) The third term is 19
b) The new term-to-term rule is subtract 13 and then divide by 6.
Step-by-step explanation:
First term: -2
Second term: 1 (-2 * 6 = -12 + 13 = 1)
Third term: 19 (1 * 6 = 6 + 13 = 19)
Our reversed sequence is now 19, 1, -2
First term: 19
Second term: 1 (19 - 13 = 6 / 6 = 1)
Third term: -2 (1 - 13 = -12 / 6 = -2)
It's term is -3 it can be divided into 6 points +79
f(x)=|3x +5| +6
g(x) = 7
Find (f + g)(x).
Answer:
(f+g)(x)
= |3x+5| + 13
Step-by-step explanation:
(f+g)(x) means
f(x) + g(x)
so you add the functions together.
|3x+5| + 6 + 7
And simplify.
|3x+5| + 13
I’m confused I need help
Answer:
x-intercept: -4
Step-by-step explanation:
It's where the slope intercepts with the x-axis
Answer:
Step-by-step explanation:
x-intercept: the x-value is that when y = 0: (-4, 0).
y-intercept: the y-value when x = 0: (0, 5)
5(2x - 12) + 21 = 2x + 41
05
08
O 10
072
Answer:
x = 10
Step-by-step explanation:
5(2x - 12) + 21 = 2x + 41 ← distribute parenthesis and simplify left side
10x - 60 + 21 = 2x + 41
10x - 39 = 2x + 41 ( subtract 2x from both sides )
8x - 39 = 41 ( add 39 to both sides )
8x = 80 ( divide both sides by 8 )
x = 10
Identify the conclusion of the conditional statement:
If an animal is a fish, then it lives in the water.
Answer:
Conclusion : it lives in the water
Step-by-step explanation:
Concept :The conclusion of a conditional statement is the result of the hypothesis. The “then” part of an if-then statement is called the conclusion.
therefore, for the given statement,
the 'then' part is 'it lives in the water'
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Ella has a collection of vintage action figures that is worth $430. If the collection
appreciates at a rate of 4% per year, which equation represents the value of the
collection after 7 years?
The equation representing the value of the collection after 7 years is value of collection = 430 + (0.04 * 430 )* 7 or value of collection = $550.4
What is an Equation ?An equation is a mathematical statement , where the algebraic expression is equated to another algebraic expression.
It is given that
Ella has a collection of vintage action figures that is worth $430.
collection
appreciates at a rate of 4% per year
equation representing the value of the collection after 7 years is
value of collection = 430 + (0.04 * 430 )* t
value of collection = 430 + (0.04 * 430 )* 7
= $550.4
Therefore the equation representing the value of the collection after 7 years is value of collection = 430 + (0.04 * 430 )* 7 or value of collection = $550.4
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What is the solution to the equation startfraction 3 over m 3 endfraction minus startfraction m over 3 minus m endfraction = startfraction m squared 9 over m squared minus 9 endfraction?
The provided equation has no solution because the square of any number cannon is negative.
What is an expression?It is defined as the combination of constants and variables with mathematical operators.
The question is incomplete.
The complete question is:
Find a solution to the given equation
Please refer to the attached picture for the expression.
We have an expression:
[tex]\rm \dfrac{3}{\left(m+3\right)}-\dfrac{3}{\left(m-3\right)}=\dfrac{m^2+9}{\left(m^2-9\right)}[/tex]
3(m - 3) - 3(m + 3) = m² + 9
3m - 9 - 3m - 9 = m² + 9
m² ≠ -27
As the square of any number cannot is negative, so no solution to the provided equation.
Thus, the provided equation has no solution because the square of any number cannon is negative.
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Answer:
D = No Solution
Step-by-step explanation:
At a gymnastics meet, twenty gymnasts compete for first, second, and third place.
How many ways can first, second, and third place be assigned
There 6840 ways can first, second, and third place be assigned.
What is Permutation?A permutation is a mathematical technique that determines the number of possible arrangements in a set when the order of the arrangements matters.
We will use the concept of permutation
[tex]^{20}P_3[/tex]= 20!/ (20-3)!
= 20*19*18*17!/ 17!
= 6840 ways.
Hence, there are 6840 ways.
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Mr Abdul made tuna and curry potato filling for some puffs.3/5 of the filling he made was tuna and the rest was curry potato.
After he used 1/8kg of the tuna filling and made another 3/4
kg of curry potato filling, he then had equal amounts of tuna filling
and curry potato filling left. How much tuna filling did he make at first?
Give your answer as a mixed number in its simplest form
At first, he made [tex]2\frac{5}{8}[/tex] kg of tuna filling.
Mr Abdul made tuna and curry potato filling for some puffs.
3/5 of the filling he made was tuna and the rest was curry potato.
So, amount of curry potato = 1- 3/5 = 2/5
Ratio of tuna & curry potato = tuna : curry potato
= 3/5 : 2/5 = 3 : 2
Let amount of tuna = 3x & amount of curry potato = 2x
After he used 1/8kg of the tuna filling and made another 3/4 kg of curry potato filling.
According to the question, he then had equal amounts of tuna filling and curry potato filling left.
So, 3x - 1/8 = 2x + 3/4
Multiplying 8 in both sides we get,
⇒ 24x - 1 = 16x +6
⇒ 24x - 16x = 7
⇒ 8x = 7
⇒ x = 7/8
So, 3x = (3×7/8) = 21/8 kg = [tex]2\frac{5}{8}[/tex] tuna filling at first.
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What is the length of a 90 degree arc of a circle with the radius of 11 cm
The axis of symmetry for the function f(x) = −x2 − 10x + 16 is x = −5. What are the coordinates of the vertex of the graph?
(−5, 41)
(−5, 56)
(−5, 76)
(−5, 91)
Answer:
the graph of f(x)= -2-10x+16
Answer:
the vertex would be (-5,41)
Step-by-step explanation:
You just plug the -5 back into the equation to get 41
If the rejection region is +1.10, what will be your decision if the z computed value is 2.34
The decision when the z value is within the rejection region is that we'll have to reject the null hypothesis.
What is a rejection region?It should be noted that the rejection region is also known as the critical region. It is the set of values where the null hypothesis is rejected.
When the observed test statistic is in the rejection region, we'll have to reject the null hypothesis and then accept rte alternative hypothesis.
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a 48 gram sample of a substance that’s used for drug research has a k value of 0.1325
Using an exponential function, it is found that the half-life of the substance is of 5.23 units of time.
What is the exponential function for the amount of a substance?The function is given by:
[tex]A(t) = A(0)e^{-kt}[/tex]
In which:
A(0) is the initial value.k is the exponential decay rate.The parameters are given by:
A(0) = 48, k = 0.1325.
Hence the equation is:
[tex]A(t) = 48e^{-0.1325t}[/tex]
The half-life is the amount of time for which A(t) = 0.5A(0) = 24, hence:
[tex]A(t) = 48e^{-0.1325t}[/tex]
[tex]24 = 48e^{-0.1325t}[/tex]
[tex]e^{-0.1325t} = 0.5[/tex]
[tex]\ln{e^{-0.1325t}} = \ln{0.5}[/tex]
[tex]0.1325t = -\ln{0.5}[/tex]
[tex]t = -\frac{\ln{0.5}}{0.1325}[/tex]
t = 5.23.
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A local mechanic on average services 14 cars a week. How many cars does he service in 4 years
if sin25=p write the following term of p
cos65
tan205
Answer:
cos65 = sin25 = p
tan205 = tan25 = p/√(1-√p)
Step-by-step explanation:
trigonometry, the branch of mathematics concerned with specific functions of angles and their application to calculations. There are six functions of an angle commonly used in trigonometry. Their names and abbreviations are sine (sin), cosine (cos), tangent (tan), cotangent (cot), secant (sec), and cosecant (cosec)
some Trigonometric identities used in the question:
cos(90-θ) = sinθsin²θ + cos²θ = 1tan(π±θ) = tanθtanθ = sinθ/cosθin the question it is given,
sin25 = p
using above mentioned identity:-
cos 65 = cos(90-25) = sin25 = p
hence value of cos65 is p.
for, tan205 we have to first find the cos25
so to find cos25 we use above mentioned identity,
cos²25 + sin²25 = 1
cos²25 + √p = 1
cos²25 = 1-√p
cos25 = √(1-√p)
now to find out tan205 use third identity mentioned above,
tan205 = tan(π+25) = tan25
tan25 = sin25/cos25
tan25 = p/√(1-√p)
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Need help urgently please
Answer:
x = -12
y = 3
Step-by-step explanation:
[tex]1/2x + 6(x+15) = 12\\1/2x + 6x + 90 = 12\\6\frac{1}{2} x + 90 = 12\\6\frac{1}{2} x= -78\\x = -12[/tex]
[tex]y = x + 15\\y = -12 + 15\\y = 3[/tex]
Find the length of side x in simplest radical form with a rational denominator.
30°
60°
√12
D
The value of x is 10√3 .
The missing figure is attached with the answer
What is a Triangle ?A triangle is a polygon with three sides , three angles and three vertices.
A right angle is a triangle in which one of the sides of a triangle is equal to 90 degree.
As it is a right angled triangle therefore the trigonometric ratios are applicable
From the given figure sin 60 needs to be determine to find the value of x
sin x = Perpendicular / Hypotenuse
sin 60 = 5 / x
[tex]\rm \dfrac{\sqrt{3}}{2} = \dfrac{5}{x}[/tex]
x = 10√3
Therefore the value of x is 10√3
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solve for z z - 6 < 3
Step-by-step explanation:
[tex]z - 6 < 3 \\ z < 3 + 6 \\ z < 9[/tex]
Hope it helps
Z – 6 < 3
Z < 3 + 6
Z < 9
✯Hope this helps✯✿..๑... Ziddi girl ♡..✧Please help! Simplify the following expression by combining like-terms.
Step-by-step explanation:
[tex] = 4x - 8y + 6 {x}^{2} + 3x - 5y[/tex]
[tex] = 6 {x}^{2} + (4x + 3x) + ( - 8y - 5y)[/tex]
[tex] = 6 {x}^{2} + 7x - 13y[/tex]
The answer is B.
Answer:
b.) 6x² + 7x - 13y
Explanation:
Given Expression
= 4x - 8y + 6x² + 3x - 5y
collect like terms
= 6x² + 4x + 3x -8y - 5y
combine similar terms
= 6x² + 7x - 13y
