The body's acceleration each time the velocity is zero is 6 [tex]m/s^{2}[/tex] or -6 [tex]m/s^{2}[/tex] and the body's speed each time the acceleration is zero is -3m/s.
According to the question,
We have the following information:
s = [tex]t^{3} -6t^{2} +9t[/tex]
Velocity = ds/dt
Velocity = [tex]3t^{2} -12t+9[/tex]
Acceleration = dv/dt
Acceleration = 6t-12
When velocity is zero:
[tex]3t^{2} -12t+9= 0[/tex]
Taking 3 as a common factor:
[tex]t^{2} -4t+3=0\\t^{2} -3t-t+3=0[/tex] (Factorizing by splitting the middle term)
t(t-3)-1(t-3) = 0
(t-3)(t-1) = 0
t = 3 or t = 1
Now, putting these values of t in acceleration's equation:
When t =3:
A = 6*3-12
A = 18-12
A = 6 [tex]m/s^{2}[/tex]
When t = 1:
A = 6*1-12
A = 6-12
A = -6 [tex]m/s^{2}[/tex]
Now, when acceleration is zero:
6t-12 = 0
6t = 12
t = 2 s
Now, putting this value in velocity's equation:
[tex]3*2^{2} -12*2+9[/tex]
3*4-24+9
12-24+9
21-24
-3 m/s
Hence, the body's acceleration each time the velocity is zero is 6 [tex]m/s^{2}[/tex] or -6 [tex]m/s^{2}[/tex] and the body's speed each time the acceleration is zero is -3m/s.
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Estimate the fraction 3/8 by rounding to the nearest whole or one-half
SOLUTION
The fraction given is
[tex]\frac{3}{8}[/tex]To round-off the fraction, we need to convert it to decimal number
[tex]\frac{3}{8}\text{ to decimal is }[/tex]Hence
The estimated fraction in decimal will be
[tex]0.375=0.4=\frac{4}{10}=\frac{2}{5}[/tex]Answer: 1/2 is the answer
hope this helped :)
Step-by-step explanation:
Write a system of equations to describe the situation below, solve using any method, and fill in the blanks.At a candy store, Carla bought 6 pounds of jelly beans and 6 pounds of gummy worms for $108. Meanwhile, Rachel bought 6 pounds of jelly beans and 1 pound of gummy worms for $63. How much does the candy cost?
Given:
Let x and y be the cost of 1 pound of Jelly beans and 1 pound of gummy worms.
[tex]\begin{gathered} 6x+6y=108 \\ x+y=18\ldots\text{ (1)} \end{gathered}[/tex][tex]6x+y=63\ldots\text{ (2)}[/tex]Subtract equation (1) from equation(2)
[tex]\begin{gathered} 6x+y-x-y=63-18 \\ 5x=45 \\ x=9 \end{gathered}[/tex]Substitute x=9 in equation(1)
[tex]\begin{gathered} 9+y=18 \\ y=18-9 \\ y=9 \end{gathered}[/tex]Cost of 1 pound of Jelly beans is $9
Cost of 1 pound of gummy worms is $9
use the first derivative test to classify the relative extrema. Write all relative extrema as ordered pairs
The given function is
[tex]f(x)=-10x^2-120x-5[/tex]First, find the first derivative of the function f(x). Use the power rule.
[tex]\begin{gathered} f^{\prime}(x)=-10\cdot2x^{2-1}-120x^{1-1}+0 \\ f^{\prime}(x)=-20x-120 \end{gathered}[/tex]Then, make it equal to zero.
[tex]-20x-120=0[/tex]Solve for x.
[tex]\begin{gathered} -20x=120 \\ x=\frac{120}{-20} \\ x=-6 \end{gathered}[/tex]This means the function has one critical value that creates two intervals.
We have to evaluate the function using two values for each interval.
Let's evaluate first for x = -7, which is inside the first interval.
[tex]f^{\prime}(-7)=-20(-7)-120=140-120=20\to+[/tex]Now evaluate for x = -5, which is inside the second interval.
[tex]f^{\prime}(7)=-20(-5)-120=100-120=-20\to-[/tex]As you can observe, the function is increasing in the first interval but decreases in the second interval. This means when x = -6, there's a maximum point.
At last, evaluate the function when x = -6 to find the y-coordinate and form the point.
[tex]\begin{gathered} f(-6)=-10(-6)^2-120(-6)-5=-10(36)+720-6 \\ f(-6)=-360+720-5=355 \end{gathered}[/tex]Therefore, we have a relative maximum point at (-6, 355).1. According to a recent poll, 4,060 out of 14,500 people in the United States were clas-sified as obese based on their body mass index. What’s the relative frequency of obe-sity according to this poll?
The relative frequency of obesity is the ratio between obese people in the sample and total people in the sample:
[tex]\frac{4060}{14500}=0.28[/tex]The relative frequency is 0.28
Tonya leaves home on her motorcycle and travels 12 miles east and 7 miles north. How far in Tonya from her original starting point?
The distance is 13.892 miles.
Given:
Distance travelled in east is 12 miles.
Distance travelled in north is 7 miles.
The objective is to find how far is tonya from the starting point.
The distance between starting point and ending point can be calculated using Pythagorean theorem.
Consider the given figure as,
By applying Pythagorean theorem,
[tex]AC^2=AB^2+BC^2[/tex]Now, substitute the given values in the above formula.
[tex]\begin{gathered} x^2=12^2+7^2 \\ x^2=144+49 \\ x^2=193 \\ x=\sqrt[]{193} \\ x=13.892 \end{gathered}[/tex]Find the missing side length and angles of ABC given that m B = 137º, a = 15, and c = 17. Round to the nearest tenth.(Find angles A and C and side b)
Given the triangle ABC and knowing that:
[tex]\begin{gathered} a=BC=15 \\ c=AB=17 \\ m\angle B=137º \end{gathered}[/tex]You need to apply:
• The Law of Sines in order to solve the exercise:
[tex]\frac{a}{sinA}=\frac{b}{sinB}=\frac{c}{sinC}[/tex]That can also be written as:
[tex]\frac{sinA}{a}=\frac{sinB}{b}=\frac{sinC}{c}[/tex]Where A, B, and C are angles and "a", "b" and "c" are sides of the triangle.
• The Law of Cosines:
[tex]b=\sqrt[]{a^2+c^2-2ac\cdot cos(B)}[/tex]Where "a", "b", and "c" are the sides of the triangle and "B" is the angle opposite side B.
Therefore, to find the length "b" you only need to substitute values into the formula of Law of Cosines and evaluate:
[tex]b=\sqrt[]{(15)^2+(17)^2-2(15)(17)\cdot cos(137\degree)}[/tex][tex]b\approx29.8[/tex]• To find the measure of angle A, you need to set up the following equation:
[tex]\begin{gathered} \frac{a}{sinA}=\frac{b}{sinB} \\ \\ \frac{15}{sinA}=\frac{29.8}{sin(137\degree)} \end{gathered}[/tex]Now you can solve for angle A. Remember to use the Inverse Trigonometric Function "Arcsine". Then:
[tex]\frac{15}{sinA}\cdot sin(137\degree)=29.8[/tex][tex]\begin{gathered} 15\cdot sin(137\degree)=29.8\cdot sinA \\ \\ \frac{15\cdot sin(137\degree)}{29.8}=sinA \end{gathered}[/tex][tex]A=\sin ^{-1}(\frac{15\cdot sin(137\degree)}{29.8})[/tex][tex]m\angle A=20.1\degree[/tex]• In order to find the measure of Angle C, you need to remember that the sum of the interior angles of a triangle is 180 degrees. Therefore:
[tex]m\angle C=180º-137º-20.1\degree[/tex]Solving the Addition, you get:
[tex]\begin{gathered} m\angle C=180º-137º-20.1\degree \\ m\angle C=22.9\degree \end{gathered}[/tex]Therefore, the answer is:
[tex]undefined[/tex]Make a estimate then divide using partial-quotients division write your remainder as a fraction
We can make an estimate for the given division by rounding the dividend to the nearest hundred and the divisor to the nearest ten.
We obtain:
[tex]812\div17\cong800\div20=40[/tex]Now, using partial-quotients division, we obtain:
17 ) 812
-170 +10 because 10*17=170
642
-170 +10
472
-170 +10
302
-170 +10
132
o from Mobile. How Tar IS Il IIUMI U 3117 12. Cincinnati is 488 miles from New York City and along the way to Brownsville, Texas which is 2007 miles from New York. How far, then, is it from Cincinnati to Brownsville?
ANSWER:
2495 miles.
STEP-BY-STEP EXPLANATION:
To answer the question we make the following scheme:
Therefore the distance between Cincinnati and Brownsville is the sum of the distance between Cincinnati to New York and New York to Brownsville, like this
[tex]\begin{gathered} d=488+2007 \\ d=2495 \end{gathered}[/tex]Therefore the distance between Cincinnati and Brownsville is 2495 miles.
You are choosing 4 of your 7 trophies and arranging them in a row on a shelfIn how many different ways can you choose and arrange the trophies?A. 840B. 28C. 24D. 5040
The formula to find how many different ways are there to choose a subgroup of r things from a group of n things is
[tex]\frac{n!}{(n-r)!}[/tex]Here, you have 7 trophies and you want to choose 4 of them, so you have
[tex]\frac{7!}{(7-4)!}\text{ = }\frac{5040}{6}=840[/tex]So there are 840 ways to choose your 4 trophies out of the 7 you have.
As a fraction in simplest terms, what would you multiply the first number by to get the second? First number: 56 Second number: 57
We're asked to find a number x such that by being multiplied by 36 becomes 57, so we need
[tex]\begin{gathered} 56x=57 \\ x=\frac{57}{56} \end{gathered}[/tex]then
[tex]56(\frac{57}{56})=57[/tex]what are the consecutive perfect cubes which added to obtain a sum of 100?441?
Answer:add 341 more cubes and that shall be your answer
1,2, 3 and 4 are the consecutive perfect cubes which added to obtain a sum of 100.
What is Number system?A number system is defined as a system of writing to express numbers.
Consecutive perfect cubes which added to obtain a sum of 100
Perfect cubes are the numbers that are the triple product of the same number.
1³+2³+3³+4³
One cube plus two ube plus three cube plus four cube
1+8+27+64
One plus eight plus twenty seven plus sixty four.
100
1,2, 3 and 4 are the consecutive perfect cubes which added to obtain a sum of 100.
Hence, 1,2, 3 and 4 are the consecutive perfect cubes which added to obtain a sum of 100.
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I tried but immediately got confused on what to start with
Radius of the inscribed circle.
Given:
Side length of square = 8cm
From the diagram, the circumference of the inscribed circle touches the sides of the square. Hence, we can say that the diameter of the inscribed circle is equal to the side length of the square.
The diagram below shows this relationship
We know that the radius (r) is related to the diameter (d) as
Write an equation for the area and solve the equation for x.
Given the figure of a rectangle
The area = A = 26
Length = x + 6
width = x + 2
Area = length * Width
so, the equation of the area will be:
[tex]A=(x+6)(x+2)[/tex]so,
[tex](x+6)(x+2)=26[/tex]solve for x as follows:
[tex]\begin{gathered} x^2+8x+12=26 \\ x^2+8x+12-26=0 \\ x^2+8x-14=0 \\ \end{gathered}[/tex]Use the general rule to find the value of x
So,
[tex]\begin{gathered} a=1,b=8,c=-14 \\ x=\frac{-b\pm\sqrt[]{b^2-4ac}}{2a}=\frac{-8\pm\sqrt[]{64-4\cdot1\cdot-14}}{2\cdot1} \\ \\ x=\frac{-8\pm\sqrt[]{120}}{2}=\frac{-8\pm2\sqrt[]{30}}{2}=-4\pm\sqrt[]{30} \end{gathered}[/tex]So, the answer will be:
[tex]\begin{gathered} A=(x+6)(x+2)_{} \\ \\ x=-4+\sqrt[]{30},-4-\sqrt[]{30} \end{gathered}[/tex]What is the inmage of A(2, 1) after reflecting it across x =4 and then across the x-axis? a.16. 1)b. (6,-1) c(-6,-1) d. -6, 1)
If we reflect the point A (2, 1) across x=4, the point will be at (6, 1). Then if we reflect it again across the x axis, the point will be at (6, -1)
Answer: (6, -1)
Consider the line y=7x-7Find the equation of the line that is perpendicular to this line and passes through the point (-8,5) Find the equation of the line that is parallel to this line and passes through the point (-8,5)
Given:
The equation of a straight line is,
[tex]y=7x-7[/tex]The objective is to find,
a) The equation of perpendicular line passes throught the point (-8,5).
b) The equation of parallel line passes throught the point (-8,5).
Explanation:
The general equation of straight line is,
[tex]y=mx+c[/tex]Here, m represents the slope of the straight line and c represents the y intercept.
a)
For perpendicular lines, the prouct of slope of two lines will be (-1).
By comparing the general equation and the given equation the slope value will be,
[tex]m_1=7[/tex]Now, the slope value of perpendicular line can be calculated as,
[tex]\begin{gathered} m_1\times m_2=-1 \\ 7\times m_2=-1 \\ m_2=-\frac{1}{7} \end{gathered}[/tex]Since, the perpendicular line passes through the point (-8,5), the equation of line can be calculated using point slope formula.
[tex]\begin{gathered} y-y_1=m_2(x-x_1)_{} \\ y-5=-\frac{1}{7}(x-(-8)) \\ y-5=-\frac{1}{7}(x+8) \\ y-5=-\frac{x}{7}-\frac{8}{7} \\ y=-\frac{x}{7}-\frac{8}{7}+5 \\ y=-\frac{x}{7}-\frac{8}{7}+\frac{35}{7} \\ y=-\frac{x}{7}+\frac{27}{7} \end{gathered}[/tex]Hence, the equation of perpendicular line is obtained.
b)
For paralle lines the slope value will be equal for both lines.
[tex]m_1=m_3=7[/tex]Since, the parallal line passes through the point (-8,5), the equation of line can be calculated using point slope formula.
[tex]\begin{gathered} y-y_1=m_3(x-x_1) \\ y-5=7(x-(-8)) \\ y-5=7(x+8) \\ y-5=7x+56 \\ y=7x+56+5 \\ y=7x+61 \end{gathered}[/tex]Hence, the equation of parallel line is obtained.
In a right triangle, cos (2x) = sin (8x + 5)'. Find the smaller of the triangle's two
acute angles.
According to the given problem,
[tex]\cos (2x)^{\circ}=\sin (8x+5)^{\circ}[/tex]Consider the formula,
[tex]\sin (90-\theta)=\cos \theta[/tex]Apply the formula,
[tex]\sin (90-2x)=\sin (8x+5)[/tex]Comparing both sides,
[tex]\begin{gathered} 90-2x=8x+5 \\ 8x+2x=90-5 \\ 10x=85 \\ x=\frac{85}{10} \\ x=8.5 \end{gathered}[/tex]Obtain the value of the two angles,
[tex]\begin{gathered} 2x=2(8.5)=17 \\ 8x+5=8(8.5)+5=73 \end{gathered}[/tex]It is evident that the smaller angle is 17 degrees, and the larger angle is 73 degrees.
Thus, the required value of the smaller acute angle of the triangle is 17 degrees.
From a point on the North Rim of the Grand Canyon, the angle of depression to a pointon the South Rim is 2. From an aerial photo, it can be determined that the horizontaldistance between the two points is 10 miles. How many vertical feet is the South Rimbelow the North Rim (nearest whole foot).
Question:
Solution:
The situation of the given problem can be drawn in the following right triangle:
This problem can be solved by applying the trigonometric identities as this:
[tex]\tan(2^{\circ})=\frac{y}{10}[/tex]solving for y, this is equivalent to:
[tex]y\text{ = 10 tan\lparen2}^{\circ}\text{\rparen=0.349 miles}[/tex]now, 0.349 miles is equivalent to 1842.72 feet. Now, this number rounded to the nearest whole foot is equivalent to 1843.
Then, the correct answer is:
1843 feet.
Please help me find the inverse of f(x) = 2^x. I think that will help me label these?
Given function:
[tex]f(x)=2^x[/tex]To obtain the inverse of the function f(x), we follow the steps outlined below:
Step 1: Replace f(x) with y:
[tex]y=2^x[/tex]Step 2: Interchange x and y
[tex]x=2^y[/tex]Step 3: Solve for y:
[tex]\begin{gathered} \text{Take logarithm of both sides} \\ \log x=log2^y \\ y\log 2\text{ = log x} \\ \text{Divide both sides by log2} \\ y\text{ = }\frac{\log x}{\log \text{ 2}} \\ y\text{ = }\log _2x \end{gathered}[/tex]Step 4: Replace y with f-1(x):
[tex]f^{-1}(x)\text{ = }\log _2x[/tex]Answer:
[tex]f^{-1}(x)\text{ = }\log _2x[/tex]The ordered pairs are graphed from the table in question 27 which of the following lines show the correct relationship
To find the price per pound of oranges, we need to divide the cost by the number of pounds:
[tex]\text{ Price / pound=}\frac{12}{3}=4\text{ \$/pound}[/tex]Thus, each pound costs $4.
The cost of 5 pounds is:
[tex]5*4=20[/tex]The answer is C. $20
I need help with this practice from my ACT prep guide onlineI’m having trouble solving it It asks to graph, if you can, use Desmos
Given:
[tex]f(x)=-4\cos (\frac{2}{3}x+\frac{\pi}{3})-3[/tex]Graph of function is cos from.
Period of the function is:
Check for period inter.
[tex]=3\pi[/tex]Writing Equations Is As Easy As 1, 2, 3 Digital Write the equation of the line that has the indicated slope and y-intercept. Slope = 2; y-intercept is (0,5)
The general structure of a linear function is "slope-intercept" form is
y=mx+b
Where
m is the slope
b is the y-intercept
To write the equation for a slope 2 and y-intercept (0,5) you have to replace said values in the formula:
m=2
b=5
y=2x+5
I need help with statistical problem I have got the answer of 0.3354 because I subtracted 0.9991 - 0.146 I wanted to know if that was correct I kept getting the answer wron
From the quetion
We are given a normal distribution with mean = 0 and standard deviation = 1
The sketch of the distribution is as shown below
Therefore option C is the correct answer
We are to find the probability that a given score is between -2.18 and 3.74
The probability is
[tex]P\mleft(-2.18Therefore,The probability is 0.9853
Given that the points (-2, 10), (5, 10), (5, 1), and (-2, 1) are vertices of a rectangle, how much longer is the length than the width? A) 1 unit B) 2 units 0) 3 units D) 4 units E) 5 units
The length of both sides is obtained by subtracting one coordinate from another sharing a similar coordinate.
(-2,10) - (5,10) = (-7,0)
These points are 7 units apart.
Let's compare the other length.
(5,10) - (5,1) = ( 0, -9)
These points are 9 units apart.
Therefore, the length is longer than the breadth by 9 - 7 = 2 units
Option B
Evaluate the expression. 2 13 21 The value of the expression is
To solve the exercise you can use the following property of powers
[tex](\frac{a}{b})^n=\frac{a^n}{b^n}[/tex]Then, you have
[tex]\begin{gathered} |(\frac{-1}{2})^3\div(\frac{1}{4})^2|=|\frac{(-1)^3}{(2)^3}^{}\div\frac{(1)^2}{(4)^2}^{}| \\ |(\frac{-1}{2})^3\div(\frac{1}{4})^2|=|\frac{-1^{}}{8}^{}\div\frac{1}{16}^{}| \end{gathered}[/tex]Now, apply the definition of fractional division, that is
[tex]\frac{a}{b}\div\frac{c}{d}=\frac{a\cdot d}{b\cdot c}[/tex][tex]\begin{gathered} |(\frac{-1}{2})^3\div(\frac{1}{4})^2|=|\frac{-1^{}\cdot16}{8\cdot1}^{}| \\ |(\frac{-1}{2})^3\div(\frac{1}{4})^2|=|\frac{-1^{}6}{8}^{}| \\ |(\frac{-1}{2})^3\div(\frac{1}{4})^2|=|-2| \end{gathered}[/tex]Finally, apply the definition of absolute value, that is, it is the distance between a number and zero. The distance between -2 and 0 is 2.
Therefore, the value of the expression is 2.
[tex]\begin{gathered} |(\frac{-1}{2})^3\div(\frac{1}{4})^2|=|-2| \\ |(\frac{-1}{2})^3\div(\frac{1}{4})^2|=2 \end{gathered}[/tex]What is the y intercept of this table?Х 0,3,6. y 5,11,17
We are given a table of x-values and their corresponding y values for a function. We are asked to express the y-intercept.
Since the table reads that for x= 0 the associated value id y = 5, then right from that info we can say that the function intercepts the y axis at the point y=5.
In coordinate pair point it reads like: (0, 5)
Recall that the y-intercept is the point at which the function crosses the y-axis, and that happens when x = 0.
Find the area of the triangle below.Be sure to include the correct unit in your answer.025 ft24 ft7 ft
To calculate the area of the triangle you have to multiply its base by its height and divide the result by 2, following the formula
[tex]A=\frac{bh}{2}[/tex]Considering the given right triangle, the base and the height of the triangle are its legs:
The area can be calculated as:
[tex]\begin{gathered} A=\frac{bh}{2} \\ A=\frac{7\cdot24}{2} \\ A=\frac{168}{2} \\ A=84ft^2 \end{gathered}[/tex]The area of the triangle is 84ft²
Carolyn has a circular swimming pool with a diameter of 20 feet. She needs to know the area of the bottom of the pool so that she can find out how much paint to buy for it. What is the approximate area?
To find the area of the bottom we have to use the formula to find the area of a circle:
[tex]A=\pi r^2[/tex]Where A is the area and r is the radius.
The first step is to find the radius of the circle, which is half the diameter:
[tex]\begin{gathered} r=\frac{D}{2} \\ r=\frac{20ft}{2} \\ r=10ft \end{gathered}[/tex]Replace r in the given formula and use 3.14 as pi:
[tex]\begin{gathered} A=3.14\cdot(10ft)^2 \\ A=314ft^2 \end{gathered}[/tex]The answer is 314ft^2.
Set B and Set C are grouped according to the Venn Diagram below. Set B is (9, 12, 14, 17, 18) and Set C is (6,9,11, 12, 18, 19). The sample space is (1, 6, 9, 11, 12, 14, 17, 18, 19, 20).
To get the probability of an event to occur, we have the following formula:
[tex]P=\frac{no.\text{ of favorable outcomes}}{\text{total no. of possible outcomes}}[/tex]According to the problem, the sample space is (1, 6, 9, 11, 12, 14, 17, 18, 19, 20) therefore, the total no. of possible outcomes is 10.
For Set B, the sample is (9, 12, 14, 17, 18), therefore, there are 5 possible outcomes that belong to set B.
Starting with the first question, what is the probability of Set B to occur?
[tex]P=\frac{no.\text{ of outcomes from B}}{\text{total no. of possible outcomes}}=\frac{5}{10}=\frac{1}{2}=0.50=50\text{ percent}[/tex]For Set C, the sample is (6,9,11, 12, 18, 19) therefore, there are 5 possible outcomes that belong to set C as well.
On the next question, what is the probability of Set C to occur?
[tex]P=\frac{no.\text{ of outcomes from C}}{\text{total no. of possible outcomes}}=\frac{5}{10}=\frac{1}{2}=0.50=50\text{ percent}[/tex]For the third question, what is the probability of Set B or C to occur?
Since the outcomes under B or C are (6, 9, 11, 12, 14, 17, 18, 19), the probability of the union of B and C is:
[tex]P=\frac{no.\text{ of outcomes from B or C}}{\text{total no. of possible outcomes}}=\frac{8}{10}=\frac{4}{5}=0.80=80\text{ percent}[/tex]On to the last question, what is the probability of the intersection of B and C to occur?
Since the outcomes that are found on both B and C are (9,12,18), the probability of the intersection of B and C is:
[tex]P=\frac{no.\text{ of outcomes found on both B and C}}{\text{total no. of possible outcomes}}=\frac{3}{10}=0.30=30\text{ percent}[/tex]Are the ratios 1:6 and 14:18 equivalent
I don't know I need points you are loved tho I don't know I need points you are loved tho
Which of the following lines is parallel to the line y= -3/2x-4?
ANSWER:
1st option: y = -3/2x + 5
STEP-BY-STEP EXPLANATION:
We have that the equation in its slope-intercept form is the following:
[tex]\begin{gathered} y=mx+b \\ \\ \text{ where m is the slope and y-intercept is b} \end{gathered}[/tex]Two lines are parallel when the slope is the same, therefore, the line parallel to this line must have a slope equal to -3/2.
We can see that option 2 is the same line, therefore, they cannot be parallel, so the correct answer is 1st option: y = -3/2x + 5