if b is the span of the given vectors, so we can express b as the linear combination of the other given two vector.
given that,
[tex]a_{1} =\left[\begin{array}{ccc}-1&\\3&\\-1\end{array}\right] , a_{2} =\left[\begin{array}{ccc}-2\\-3\\6\end{array}\right] , b=\left[\begin{array}{ccc}-6\\9\\2\end{array}\right] \\\\\\\\\\\\\\\\\\\\\\\\[/tex]
We have determined that b is span of given vector a1 and a2.
we can write. [tex]a_{1} =(-1,-3,-1) ,a_{2}=(-2,-3,6) , b=(-6,9,2)[/tex] .
A vector is a quantity that has magnitude and direction.
if x,y,z...., v are vectors and a,b,c....k are scalars then the linear combination is ax+by+cz+......+kv
The set of linear combinations is called span for set of vectors.
So, b is span of given vector.
[tex]b=(c)a_{1} +(d)a_{2}[/tex] ...... equation (A)
where c and d are scalars.
Now,
[tex](-6,9,2)=c(-1,3,-1)+d(-2,-3,6)\\(-6,9,2)=(-c,3c,-c)+(-2d,-3d,6d)\\(-6,9,2)=(-c-2d,3c-3d,-c+6d)\\[/tex]
equating the equations,
[tex]-c-2d=-6\\3c-3d=9\\-c+6d=2\\[/tex]
now we shall solve the above linear equations.[tex]subtract \\-c-2d-(-c+6d) = -6-2\\-8d=-8\\d=1\\now, \-c=-6+2d\\c=6-2=4\\c=4[/tex]
So, value of scalars c, d satisfied equation A,
thus, b is span of given vector a1 and a2.
The complete question is.
Determine if b is in the span of the other given vectors. If so, write b as a linear combination of the other vectors. a1=[-1 3 -1], a2=[-2 -3 6]= [-6 9 2]
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the graphs below show changes in a population over time. the dashed line represents the carrying capacity of the species. which of the following graphs best shows the population size of a k-selected species over time in a stable environment?
The graph in the upper left corner best shows the population size of a K-selected species over time in a stable environment.
This graph shows the population of the species increasing at a slow rate until it reaches the carrying capacity of the species (the dashed line). After reaching the carrying capacity, the population of the species remains relatively stable, fluctuating slightly around the carrying capacity. This is evidence of a K-selected species, which tend to have a higher carrying capacity and longer life spans, and the population size remains relatively stable over time in a stable environment. Mathematically, this can be expressed as y = K = C, where y is the population size, K is the carrying capacity, and C is the constant value of the carrying capacity.
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Can someone help me pleace
Answer:
A
Step-by-step explanation:
ABC is similar to DEF because similar triangles are triangles that have the same shape, but their sizes may vary.
7-89. For angle α in the first quadrant, . Use that information to find each of the
following values without using a calculator. Be prepared to share your strategies with the class.
a. sin α
b. sin(π + α)
c. cos(2π − α)
The required values are for the given angle, as follows: a. sin α is positive. b. sin(π + α) = -sin α. c. cos(2π - α) = cos α.
What is the quadrant?A quadrant is defined as an area contained by the x and y axes, which there are four quadrants in a graph.
a. For an angle α in the first quadrant, sin α is positive.
Since the first quadrant is between 0 and 90 degrees, sin α is equal to the opposite side divided by the hypotenuse, where the opposite side is the y-coordinate and the hypotenuse is the distance from the origin to the point on the unit circle corresponding to the angle α.
b. sin(π + α) = sin (π - α). In the second quadrant (90 < α < 180), sine is negative, so sin(π + α) = -sin α.
c. cos(2π - α) = cos α. The cosine function has period 2π, so cos(2π - α) = cos α.
Since the first quadrant is between 0 and 90 degrees, cos α is positive.
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THREE PARTS QUESTION PLEASE HELP EASYY
Philip pours wax into a cylindrical mold to make candles. The dimensions of a finished candle are shown in the diagram below.
PART
A candle shaped like a cylinder is shown. The height of the candle is labeled five and five tenths inches. A dashed line is drawn from the center of the circular base to the edge. The line is labeled three inches.
Question 1
Part A
How much wax does Philip use to make a candle rounded to the nearest cubic inch? Use 3.14 for π.
Enter the correct answer in the box.
Question 2
Part B
Philip spends $0.15 per cubic inch for the wax to make the candles. Approximately how much will it cost him to purchase enough wax to make 14 cylindrical candles?
A) $109.20 dollars
B) $218.40 dollars
C) $325.50 dollars
D) $598.50 dollars
Question 3
Part C
Philip wants to cover the lateral surface of one of the candles with glitter. How many square inches of the candle will he cover with glitter in terms of π?
Enter the correct answer in the box in terms of π.
Part A
Using the formula for the volume of a cylinder,
[tex]V=\pi r^2 h \approx (3.14)(3^2)(5.5)=\boxed{155.43}[/tex]
Part B
The total volume of 14 candles is [tex](155.43)(14)=2176.02[/tex] cubic inches.
So, the cost is [tex](2176.02)(0.15) \approx \boxed{\$325.50}[/tex]
Part C
Using the formula for the lateral surface area of a cylinder,
[tex]2\pi rh=2\pi(3)(5.5)=\boxed{33\pi}[/tex]
PLEASE HELP ASAP (pic included)
Find the
equation
of this
line.
y = [? ]x + [ ]
Simplify your answer and enter as an integer.
Answer:
y=-3x+4
Step-by-step explanation:
If we use rise over run from one point to the next, you will see that you rise vertically 3 units, then run horizontally 1 unit.
Then Make the rise negative because this is a negative graph. So -3/1
Divide -3 by 1. -3/1=-3. mx=-3
Now we have to find the y-intercept. This is where the graph crosses/intersects the y-axis. As if you look at the y-axis, you will see that the graph intersects the y-axis at (0,4).
So your y-intercept is 4. b=4.
-Hope this helped
a wall 10 m long 1m high and 1/2 m wide is to be made of bricks . The bricks are of size 10 multiply 10 multiply 20 . how many bricks will be needed to make the wall.
To find out how many bricks are needed to make the wall, we need to find the total area of the wall and divide it by the area of one brick.
The area of the wall is:
length x height x width = 10 m x 1 m x 0.5 m = 5 m^2
The area of one brick is:
10 cm x 10 cm x 20 cm = 2000 cm^3
To convert the area of the wall from square meter to cubic centimeter, we need to multiply it by 100*100 = 10000.
5 m^2 x 10000 = 50000 cm^2
Now we can divide the total area of the wall by the area of one brick to find out how many bricks are needed:
50000 cm^2 / 2000 cm^2/ brick = 25 bricks
So, 25 bricks will be needed to make the wall.
Solve for x
8√3
12.
Select the correct response:
8
16
15
4√3
X
Answer:
see below & attached
Step-by-step explanation:
red line length call it a² = (8√3)²- 12² = 192-144 =48
so look at the graph we know x is longer than 12,
the portion that's longer than 12 call it c = x-12
top short line call it b
because it's all right angle you are looking at 3 right angle triangles
the big one that's cut into 2 by the red line
the larger one & the smaller one
each will have this property: short side 1 ²+short side 2 ²= long side ²
so for the largest one:
(8√3)² +b² = (12+c)² = x²
second largest triangle we already know all 3 sides: (8√3), 12, √48
now look at the smallest
48+c² =b²
once you have all equations listed , you can solve for c.
see attached.
The sum of four consecutive odd integers is 264. Find the integers.
The smallest of the four consecutive integers is?
Answer:
63
Step-by-step explanation:
Consecutive odd integers increase by the addition of 2 therefore if the smallest value is x then the next odd number is x + 2
So therefore; x + x + 2 + x + 4 + x + 6 = 264
Collect like terms
x + x + x + x + 2 + 4 + 6 = 264
4x + 12 = 264
4x = 264 - 12
4x = 252
Divide both sides by 4
x = 63
Therefore the smallest of the four consecutive integers is 63
Explain why 96:38 is NOT the same as 38:96.
The two ratios are inverse of each other that's why 96:38 is not the same as 38:96.
What are inverse ratios?When two quantities are inversely related, that is, when an increase in one quantity causes a decrease in the other and vice versa, they are said to be in inverse proportion. The product of the given two quantities is equal to a constant value in inverse proportion.
Given that the two ratios are 96:38 and 38:96. The value of the two ratios will not be the same because both the ratios are inverse of each other or are reciprocals of each other.
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Answer:
The two ratios are inverse of each other that's why 96:38 is not the same as 38:96.
Step-by-step explanation:
I took the quiz.
I just need to figure out how to write the equation
The rest I can do on my own
Katie is mixing a new blend of perfume. Her formula calls for 12% lavender solution. She has two bottles. One is 15% and the other is 6%. How many ounces of each does she need to make 18 ounces of 12%?
On solving the equation, it is seen that about 12 ounces of the 15% solution must be mixed with about 6 ounces of the 6% solution to get 18 ounces of the 12% perfume.
What is an equation?
A mathematical definition of an equation is a claim that two expressions are equal when they are joined by the equals sign ("=").
Let x = number of liters of the 15% solution
Let 18 − x = number of liters of the 6% solution
Solution Ounces Strength Amount of Perfume
15% x 15 15x
6% 18 - x 6 6(18 - x)
Mixture 18 12 (18)(12)
So, the equation will be -
(amount of perfume in 15% solution) + (amount of perfume in 6% solution) = (amount of perfume in mixture)
15x + 6(18 − x) = (18)(12)
Solve the equation using distributive property -
15x + 108 - 6x = 216
Combine the like terms -
15x - 6x = 216 - 108
Simplify the equation further -
9x = 108
x = 108/9
x = 12
Now, substituting the value of x in 18 − x = 18 - 12 = 6.
Therefore, the amount in ounces is obtained as 12 ounces and 6 ounces.
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One federal tax schedule states that if you are single and your taxable income is between $37,450 and $90,750, then you owe $5156.25 plus 25%
of your taxable income over $37,450. Is the tax you owe a linear function of your taxable income if the taxable income is between $37,450 and $90,750?
If so, find a formula giving tax owed as a linear function of taxable income over $37,450.
The linear function for the amount of tax owed for a taxable income over $37,450 is given as follows:
y = 0.25x + 4191.5.
How to define the linear function?The slope-intercept definition of a linear function is given as follows:
y = mx + b.
In which the parameters are given as follows:
m is the slope.b is the intercept.If your taxable income is between $37,450 and $90,750, then you owe $5156.25 plus 25% of your taxable income over $37,450, hence the function is defined as follows:
y = 5156.25 + 0.25(x - 37450).
y = 5126.5 + 0.25x - 935
y = 0.25x + 4191.5.
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please i need HELPPP
Answer:
vertex point is (4,-6)
and it opens upwards as the coefficient of x is positive.
I assume B is the correct answer, good luck!
What is the size, in degrees, of the missing angle?
45°
65%
The measure of the missing angle in a triangle is C = 18°.
What is a 45-Degree Angle?An acute angle is one that is 45 degrees. It is a 90-degree angle or half of a right angle. Look at the illustration of the 45-degree angle. Angles are expressed in degrees or radians.
A straight angle is one in which the two arms extend in precise opposition to one another. A 180° angle is a straight angle. A protractor can be used to measure angles; a straight angle is an angle that is measured at 90 degrees.
The two arms are perpendicular to one another at a right angle. Each angle measures 45° when the right angle is divided into two equal pieces. An acute angle is one that is 45 degrees. It is a 90-degree angle or half of a right angle.
Given one of the angles A = 45°
And B is 65% of the total angle sum = 65/100 × 180
= 117°
For angle C, using angle sum property
⇒ A + B + C = 180°
⇒ 45° + 117° + C = 180°
⇒ C = 180° - (45° + 117°)
⇒ C = 180° - 162°
⇒ C = 18°
Thus, The measure of the missing angle in a triangle is C = 18°.
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Complete question:
What is the size, in degrees, of the missing angle of a triangle?
A = 45°
B = 65%
C = x
Determine if the following
lengths below will create a triangle
(3,4,5)
(10, 20, 30)
(15.5, 45, 20.5)
a. (3, 4, 5) will create a triangle.
b. (10, 20, 30) will not create a triangle
c. (15.5, 45, 20.5) will not create a triangle.
What is a triangle?
A triangle is a polygon with three sides and three angles. The sum of the interior angles of a triangle is always 180 degrees. Triangles can be classified by their sides (such as equilateral, isosceles, or scalene) or by their angles (such as acute, right, or obtuse).
The set of lengths (3,4,5) will create a triangle. This is because it follows the "triangle inequality theorem" which states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
The set of lengths (10, 20, 30) will not create a triangle. This is because 10 + 20 is less than 30, so it does not satisfy the triangle inequality theorem.
The set of lengths (15.5, 45, 20.5) will not create a triangle. This is because 15.5 + 20.5 is less than 45, so it does not satisfy the triangle inequality theorem.
Hence,
a. (3, 4, 5) will create a triangle.
b. (10, 20, 30) will not create a triangle
c. (15.5, 45, 20.5) will not create a triangle.
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two sides of a triangle are 19.75 and 10.25. select all choices that could represent the third side of the triangle.
the two possible values for the third side of the triangle are 22.5 and 24.00.
We know that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
Therefore, the third side must be greater than 10.25 and less than 19.75.
From this, we can conclude that the third side of the triangle must be either 22.5 or 24.00.
We know that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. Given that two sides of the triangle measure 19.75 and 10.25, the third side must be greater than 10.25 and less than 19.75. Therefore, the only two possible lengths for the third side of the triangle are 22.5 and 24.00. Any length greater than 19.75 or less than 10.25 would not satisfy the triangle inequality and would not be a valid third side for the triangle. Thus, We know that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
the two possible values for the third side of the triangle are 22.5 and 24.00.
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the inside of a flashlight surrounding the bulb is shaped like a parabola in which the base of the bulb represents the vertex and the top of the bulb represents the focus. which of the following standard equations represents the parabola in which the base of the bulb is at (1, 5) and top of the bulb is at (1, 1)? (x 1)2
The equation for the parabola is: (x - 1)^2 = 4(y - 5). The vertex of the parabola is located at (1, 5) and the focus is at (1, 1). The equation describes a parabola that opens upward, and is symmetrical about the y-axis.
The equation for the parabola is: (x - 1)^2 = 4(y - 5). This equation describes a parabola with its vertex located at (1, 5) and its focus at (1, 1). The parabola is symmetrical about the y-axis and opens upward. This means that the base of the bulb is at (1, 5) and the top of the bulb is at (1, 1). To get the equation for the parabola, we need to subtract the x-coordinate of the vertex from the x-coordinate of the focus, which gives us 1. We then square this number, giving us 1. We then subtract the y-coordinate of the vertex from the y-coordinate of the focus, giving us 4. We then multiply this number by the squared number from before, giving us 4. The final equation is then (x - 1)^2 = 4(y - 5).
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How many different committees can be formed from 11 teachers and 50 students if the committee consists of 2 teachers and 2 students?
Answer:
5 committees can be formed
Step-by-step explanation:
number of teachers=11
number of students=50
number of teachers in a committee=2
number of students in a committee=2
Dave has inherited an apple orchard on which 70 trees are planted. Under these conditions, each tree yields 10 bushels of apples. According to the local WSU extension agent, each time Dave removes a tree the yield per tree will go up 0.45 bushels. Let x be the number of trees in the orchard and N the yield per tree. (a) Find a formula for N in terms of the unknown x. (Hint: Make a table of data with one column representing various values of x and the other column the corresponding values of N. After you complete the first few rows of the table, you need to discover the pattern.) 41.5 – 0.45x x (b) What possible reason(s) might explain why the yield goes up when you remove trees? removing trees will mean that you will be able to share more nutrients for the trees that are left, which produces more apples per tree, and better quality of the apples as well. The yield increases as the quality and care increases for the trees.
X = 88.1. The total of all interest, dividends, or other income that an investment provides divided by the age of the investment or the period over which the investor has held it is the average yield on an investment.
What is average yield?
The average yield on an investment or a portfolio is calculated by dividing the total interest, dividends, or other income received by an investment by the investment's age or the period over which the investor has held it. In particular, it alludes to yield, or the total revenue from an investment divided by the number of years kept.
There are numerous yield kinds, and the calculating process varies depending on the specific yield type and the type of security. Because of these variations, comparisons of yields between other financial product categories should be used with caution.
In the case of stocks, yield is determined by dividing the price rise plus dividends by the original purchase price. Both cost yield and current yield can be used to examine bond yield.
Given data :
Let x be the number of trees in the orchard and N the yield per tree.
No of trees = 70
yield = 10
formula for N in terms of the unknown x :
(10+(70-x)0.55)
10 + 38.5 - 0.55x
-0.55x = 48.5
x = 48.5/ 0.55 = 88.1
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Find the least common denominator of 1/2x-6 and 2x/7x-21.
The least common denominator to both fraction expressions is; (x - 3)
How to find the least common denominator?The least common denominator simply means the least number or expression that is common to both denominators of a given fraction.
Now, the given denominators of the fraction expression are;
(2x - 6) and (7x - 21)
To find the least common denominator here, we will just factoriuze each algebraic expression to getl
(2x - 6) = 2(x - 3)
(7x - 21) = 7(x - 3)
Thus, we see that the common expression is (x - 3) which we will conclude will be the least common denominator
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SEE IMAGE!!! PLS HELP ASAPPP
Find the quotient. If possible, rename the quotient as a mixed number. Select the answer that is in simplest form.
3 5
8 ÷ 4 = ?
A.
1 5
8
B.
29
32
C.
25
32
Answer: B
Step-by-step explanation:
[tex]3 \frac{5}{8} \div 4=\frac{29}{8} \div 4=\frac{29}{32}[/tex]
5+w=8 simplified in to simplest form
[tex] \large \frak{ \green 5 + \red{w }= \green 8}[/tex]
[tex] \: [/tex]
[tex] \ \: \large \frak{ \red{w }= \green 8 { -} \green5}[/tex]
[tex] \: [/tex]
[tex] \underline{ \boxed{ \large \frak{ \: \red{w }= \green 3 \: }}}[/tex]
[tex] \: [/tex]
hope it helps!
Find the volume and the total area of the largest cube of the wood that can be from a log of circular section whose radius is 16.8in.
The volume and the total area of the largest cube of the wood that can be from a log of circular section whose radius is 16.8in is: 7.76 ft³, 23.52 ft².
How to find the volume and total area?Diameter of the log
d=2(16.8)
d=33.6in
Edge of cube
a²+a²=d²
2a²=33.6²
a²=564.48
a=23.76 in
Volume of the largest cube
V=a³
V=23.76³
V=13413.41 in³
V=13413.41 in³×(1 ft/12 in)³
V=7.76 ft³
Total area of the largest cube
A=6a²
A=6(23.76²)
A=3,387.23in²
A=3,387.23 in²×(1/ ft12 in)²
A=23.52 ft²
Therefore the Volume of the largest cube is 7.76 ft³ and the total area is 23.52 ft².
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Si mi tía llega a la casa con cuatro donas de azúcar y estamos seis personas en la casa ¿ de cuánto nos tocaría ha cada uno en fracciones?
A coda ulna de las personas le Toccara ulna porcino de 2/3 de Dona
de canto no's Toccara ha coda uno eon fractions?Si mi tía ha legato a la casa y trae consigo 4 Donas de azúcar, pero no se ha percatado que la cantidad de personas en la casa es de 6 contándose, entonces decide repartir en partes iguales, para esta operation deems divider la candida de donnas entre la cantidad de personas
4 donnas / 6 personas = 2 /3 ester 2/3 es la porcino de coda uno es décor coda Dona se picara eon tress torsos y dos torsos para coda persona de 2/3
Respuesta:
para que alkanes les tocaria
a 4 personas la mated de una dona
(osea 2 donas las partes a la mated)
y a las okras 2 personas la Dona entera
Explication paso a paso:
esparto hiver aoudad:)
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A rancher wants to enclose a rectangular field with 220 ft of fencing. One side is a river and will not require a fence. What is the maximum area that can be enclosed?
5329 Sq meters is the maximum area that can be enclosed.
What is a rectangle, exactly?
A rectangle is a sort of quadrilateral with parallel sides that are equal to one another and four vertices that are all 90 degrees apart. It is also known as an equiangular quadrilateral for this reason.
The term "parallelogram" can also be used to describe a rectangle because the opposing sides are equal and parallel.
Perimeter of Rectangle: 2W + 2L
One L is the river so we have 3 sides
220m/3 = 73m
Area of Rectangle is WL
73 × 73 = 5329 Sq meters
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1. Determine whether the following numbers could be the sides of a right triangle. Show your work.
a.
17, 15, 8
b.
7, 25, 19
a. 17, 15, 8 this can be a right triangle, while b. 7 25 19 is not a right a triangle
How to know when triangles are formedThere are an endless number of triangles produced when no side is specified.
The three angles produce an unlimited number of distinct triangles if side lengths are not provided.
When the three sides are specified the sides will be such that addition of the two smaller sides will be greater than the third side
For a. 17, 15, 8
the two smaller sides are:15 and 8
15 + 8 > 17
this can be a triangle
For b. 7, 25, 19
the two smaller sides are:7 and 19
7 + 19 > 25
this can be a triangle
For a right triangle the Pythagoras rule should be observe: the squire of the smaller sides will be equal to the square of the longer side. let the longer side be x
x² = 15² + 8²
x² = 225 + 64
x² = 289
x = √289
x = 17
this is a right triangle
x² = 7² + 19²
x ² = 49 + 361
x² = 410
x = √410
x = 20.25
this is not a right triangle
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I need help with this
The point (2, 48) means in this context: Cesar and Camden are both 48 kilometers away from home after 2 hours.
What is the system of equations?One or many equations having the same number of unknowns that can be solved simultaneously called as simultaneous equation. And simultaneous equation is the system of equation.
Given:
Cesar and Camden are brothers and live at the same house.
Cesar is 80 kilometers away from home and bikes towards home at a rate of 16 kilometers per hour.
Camden is 20 kilometers away from home and keeps walking away from home at a rate of 14 kilometers per hour.
You found that the point of intersection on the graph is (2, 48),
and you solved algebraically to find that x=2 and y = 48.
Cesar and Camden are both 48 kilometers away from home after 2 hours.
Therefore, Cesar and Camden are both 48 kilometers away from home after 2 hours.
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Write a two-column proof
given AX = DX, XB=XC
prove AC = BD
please help, this is for my midterm
In the parallelogram ABCD it is proved that AC ≅ BD, which proves that AC = BD.
What is a parallelogram?
A quadrilateral with two sets of parallel sides is referred to as a parallelogram. In a parallelogram, the opposing sides are of equal length, and the opposing angles are of equal size. Additionally, the interior angles that are additional to the transversal on the same side.
It is given that the parallelogram is ABCD.
In this parallelogram AX ≅ DX, XB ≅ XC.
Now, according to the properties of parallelogram the segment addition can be done.
AC = AX + XC
Similarly, the segment addition for line BD is -
BD = DX + BX
Now if AX ≅ DX and XB ≅ XC, then substitution of DX and XC can be done -
AC = AX + BX
BD = AX + BX
This proves that AC = BD by the transitive or substitution method.
Therefore, it is proved that AC = BD and also that AC ≅ BD.
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construct the general solution of involving complex eigenfunctions and then obtain the general real solution. describe the shapes of typical trajectories.
The ODE's general solution has real solution and complex eigen functions. Typical trajectories are sinusoidal curves.
The general solution of the ODE involves complex eigenfunctions, which are solutions of the form e^(ikx), where k is a real number and i is the imaginary unit. By combining these complex eigenfunctions with real coefficients, a general real solution can be obtained. Typically, these solutions will take the form of sinusoidal curves, with the amplitude and period determined by the coefficients of the eigenfunctions. The coefficients can be determined using initial conditions, allowing for more specific trajectories to be determined. For example, if the initial condition is known, then the coefficients of the eigenfunctions can be determined using Fourier analysis, allowing for the trajectory of the system to be determined.
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Write the other side of this equation so it’a true for no values of x: 1/2(6x - 10) -x =
Answer:
2x + 3
Step-by-step explanation:
1/2(6x - 10) - x
3x -5 -x
2x -5
We would need the structure of the equation to be 2x plus or minus anything other than 15.
For example:
2x - 5 = 2x + 3 if we subtract 2x from both sides we are left with
-5 = 3 This is a false statement and their will be no solution.
You are given the 2012 value of a product and the rate at which the value is expected to change during the next 5 years. Use this information equation that gives the dollar value V of the product in terms of the year. (Let t-12 represent 2012.) 2012 Value $159Rate $4.50 (increase per year) _____, 12 ≤ t ≤ 17
The linear equation is [tex]$V=21.45+5.15 t$[/tex].
What is the Slope-Intercept Form of a Linear Equation?When the slope of the line being studied is known, and the provided point is also the y intercept, the slope intercept formula, y = mx + b, is utilized (0, b). The y value of the y intercept point is represented by b in the equation. A line's equation can be written in the slope-intercept form such that the slope (steepness) and y-intercept (where the line crosses the vertical y-axis) are instantly visible. A common name for this form is the y = mx + b form.We are given the dollar value [tex]$\$ 72.95$[/tex] of a product in 2010 and it is changed [tex]$\$ 5.15$[/tex] increase per year.
We are asked to write a linear equation that gives the dollar value V of the product in terms of the year t.
The general linear equation for this case would be,
[tex]$$V=C+5.15 t$$[/tex]
Since the dollar value increases by [tex]$\$ 5.15$[/tex]per year, the slope is[tex]$5.15$[/tex]. And, C is the V-intercept (the value of V when t=0 ).
Let us first determine the value of $C$ by substituting the given dollar value $ 72.95 in 2010 . This gives us:
[tex]V & =C+5.15 t \\[/tex]
[tex]72.95 & =C+5.15(10) \\[/tex]
[tex]72.95 & =C+51.5 \\[/tex]
[tex]72.95-51.5 & =C+51.5-51.5 \\[/tex]
[tex]21.45 & =C[/tex]
[Given that[tex]$\mathrm{t}=10$[/tex] at 2010
[Subtract 51.5 to both sides]
Thus, the value of C=21.45.
Let us substitute the value of C in the Eqn(1):
[tex]$$V=21.45+5.15 t$$[/tex]
Therefore, the linear equation is [tex]$V=21.45+5.15 t$[/tex].
The complete question is,
You are given the dollar value of a product in 2010 and the rate at which the value of the product is expected to change during the next 5 years. Use this information to write a linear equation that gives the dollar value V of the product in terms of the year t. (Let t=10 represent 2010 .)
[tex]$\frac{2010 \text { Value }}{\$ 72.95}$[/tex]
Rate
[tex]$\$ 5.15$[/tex] increase per year
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