# 4.3 Graph with intercepts  (Page 2/5)

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## Find the x - and y - intercepts from the equation of a line

Use the equation of the line. To find:

• the x - intercept of the line, let $y=0$ and solve for $x$ .
• the y - intercept of the line, let $x=0$ and solve for $y$ .

Find the intercepts of $2x+y=6$ .

## Solution

We will let $y=0$ to find the x - intercept, and let $x=0$ to find the y - intercept. We will fill in the table, which reminds us of what we need to find.

To find the x - intercept, let $y=0$ .

 Let y = 0. Simplify. The x -intercept is (3, 0) To find the y -intercept, let x = 0. Let x = 0. Simplify. The y -intercept is (0, 6)

The intercepts are the points $\left(3,0\right)$ and $\left(0,6\right)$ as shown in [link] .

 $2x+y=6$ $x$ $y$ 3 0 0 6

Find the intercepts of $3x+y=12.$

x - intercept: $\left(4,0\right)$ , y - intercept: $\left(0,12\right)$

Find the intercepts of $x+4y=8.$

x - intercept: $\left(8,0\right)$ , y - intercept: $\left(0,2\right)$

Find the intercepts of $4x–3y=12$ .

## Solution

 To find the x -intercept, let y = 0. Let y = 0. Simplify. The x -intercept is (3, 0) To find the y -intercept, let x = 0. Let x = 0. Simplify. The y -intercept is (0, −4)

The intercepts are the points (3, 0) and (0, −4) as shown in [link] .

 $4x-3y=12$ $x$ $y$ 3 0 0 $-4$

Find the intercepts of $3x–4y=12.$

x - intercept: $\left(4,0\right)$ , y - intercept: $\left(0,-3\right)$

Find the intercepts of $2x–4y=8.$

x - intercept: $\left(4,0\right)$ , y - intercept: $\left(0,-2\right)$

## Graph a line using the intercepts

To graph a linear equation by plotting points, you need to find three points whose coordinates are solutions to the equation. You can use the x - and y - intercepts as two of your three points. Find the intercepts, and then find a third point to ensure accuracy. Make sure the points line up—then draw the line. This method is often the quickest way to graph a line.

## How to graph a line using intercepts

Graph $–x+2y=6$ using the intercepts.

## Solution

Graph $x–2y=4$ using the intercepts.

Graph $–x+3y=6$ using the intercepts.

The steps to graph a linear equation using the intercepts are summarized below.

## Graph a linear equation using the intercepts.

1. Find the x - and y - intercepts of the line.
• Let $y=0$ and solve for $x$
• Let $x=0$ and solve for $y$ .
2. Find a third solution to the equation.
3. Plot the three points and check that they line up.
4. Draw the line.

Graph $4x–3y=12$ using the intercepts.

## Solution

Find the intercepts and a third point.

We list the points in [link] and show the graph below.

 $4x-3y=12$ $x$ $y$ $\left(x,y\right)$ 3 0 $\left(3,0\right)$ 0 $-4$ $\left(0,-4\right)$ 6 4 $\left(6,4\right)$

Graph $5x–2y=10$ using the intercepts.

Graph $3x–4y=12$ using the intercepts.

Graph $y=5x$ using the intercepts.

## Solution

This line has only one intercept. It is the point $\left(0,0\right)$ .

To ensure accuracy we need to plot three points. Since the x - and y - intercepts are the same point, we need two more points to graph the line.

 $y=5x$ $x$ $y$ $\left(x,y\right)$ 0 0 $\left(0,0\right)$ 1 5 $\left(1,5\right)$ $-1$ $-5$ $\left(-1,-5\right)$

Plot the three points, check that they line up, and draw the line.

Graph $y=4x$ using the intercepts.

Graph $y=\text{−}x$ the intercepts.

## Key concepts

• Find the x - and y - Intercepts from the Equation of a Line
• Use the equation of the line to find the x - intercept of the line, let $y=0$ and solve for x .
• Use the equation of the line to find the y - intercept of the line, let $x=0$ and solve for y .
• Graph a Linear Equation using the Intercepts
1. Find the x - and y - intercepts of the line.
Let $y=0$ and solve for x .
Let $x=0$ and solve for y .
2. Find a third solution to the equation.
3. Plot the three points and then check that they line up.
4. Draw the line.

• Strategy for Choosing the Most Convenient Method to Graph a Line:
• Consider the form of the equation.
• If it only has one variable, it is a vertical or horizontal line.
$x=a$ is a vertical line passing through the x - axis at $a$
$y=b$ is a horizontal line passing through the y - axis at $b$ .
• If y is isolated on one side of the equation, graph by plotting points.
• Choose any three values for x and then solve for the corresponding y - values.
• If the equation is of the form $ax+by=c$ , find the intercepts. Find the x - and y - intercepts and then a third point.

## Practice makes perfect

Identify the x - and y - Intercepts on a Graph

In the following exercises, find the x - and y - intercepts on each graph.

$\left(3,0\right),\left(0,3\right)$

$\left(5,0\right),\left(0,-5\right)$

$\left(-2,0\right),\left(0,-2\right)$

$\left(-1,0\right),\left(0,1\right)$

$\left(6,0\right),\left(0,3\right)$

$\left(0,0\right)$

Find the x - and y - Intercepts from an Equation of a Line

In the following exercises, find the intercepts for each equation.

$x+y=4$

$\left(4,0\right),\left(0,4\right)$

$x+y=3$

$x+y=-2$

$\left(-2,0\right),\left(0,-2\right)$

$x+y=-5$

$x–y=5$

$\left(5,0\right),\left(0,-5\right)$

$x–y=1$

$x–y=-3$

$\left(-3,0\right),\phantom{\rule{0.2em}{0ex}}\text{}\phantom{\rule{0.2em}{0ex}}\left(0,3\right)$

$x–y=-4$

$x+2y=8$

$\left(8,0\right),\left(0,4\right)$

$x+2y=10$

$3x+y=6$

$\left(2,0\right),\left(0,6\right)$

$3x+y=9$

$x–3y=12$

$\left(12,0\right),\left(0,-4\right)$

$x–2y=8$

$4x–y=8$

$\left(2,0\right),\left(0,-8\right)$

$5x–y=5$

$2x+5y=10$

$\left(5,0\right),\left(0,2\right)$

$2x+3y=6$

$3x–2y=12$

$\left(4,0\right),\left(0,-6\right)$

$3x–5y=30$

$y=\frac{1}{3}x+1$

$\left(3,0\right),\left(0,-1\right)$

$y=\frac{1}{4}x-1$

$y=\frac{1}{5}x+2$

$\left(-10,0\right),\left(0,2\right)$

$y=\frac{1}{3}x+4$

$y=3x$

$\left(0,0\right)$

$y=-2x$

$y=-4x$

$\left(0,0\right)$

$y=5x$

Graph a Line Using the Intercepts

In the following exercises, graph using the intercepts.

$–x+5y=10$

$–x+4y=8$

$x+2y=4$

$x+2y=6$

$x+y=2$

$x+y=5$

$x+y=-3$

$x+y=-1$

$x–y=1$

$x–y=2$

$x–y=-4$

$x–y=-3$

$4x+y=4$

$3x+y=3$

$2x+4y=12$

$3x+2y=12$

$3x–2y=6$

$5x–2y=10$

$2x–5y=-20$

$3x–4y=-12$

$3x–y=-6$

$2x–y=-8$

$y=-2x$

$y=-4x$

$y=x$

$y=3x$

## Everyday math

Road trip. Damien is driving from Chicago to Denver, a distance of 1000 miles. The x - axis on the graph below shows the time in hours since Damien left Chicago. The y - axis represents the distance he has left to drive.

1. Find the x - and y - intercepts.
2. Explain what the x - and y - intercepts mean for Damien.

$\left(0,1000\right),\left(15,0\right)$
At $\left(0,1000\right)$ , he has been gone 0 hours and has 1000 miles left. At $\left(15,0\right)$ , he has been gone 15 hours and has 0 miles left to go.

Road trip. Ozzie filled up the gas tank of his truck and headed out on a road trip. The x - axis on the graph below shows the number of miles Ozzie drove since filling up. The y - axis represents the number of gallons of gas in the truck’s gas tank.

1. Find the x - and y - intercepts.
2. Explain what the x - and y - intercepts mean for Ozzie.

## Writing exercises

How do you find the x - intercept of the graph of $3x–2y=6$ ?

Do you prefer to use the method of plotting points or the method using the intercepts to graph the equation $4x+y=-4$ ? Why?

Do you prefer to use the method of plotting points or the method using the intercepts to graph the equation $y=\frac{2}{3}x-2$ ? Why?

Do you prefer to use the method of plotting points or the method using the intercepts to graph the equation $y=6$ ? Why?

## Self check

After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

What does this checklist tell you about your mastery of this section? What steps will you take to improve?

Larry and Tom were standing next to each other in the backyard when Tom challenged Larry to guess how tall he was. Larry knew his own height is 6.5 feet and when they measured their shadows, Larry’s shadow was 8 feet and Tom’s was 7.75 feet long. What is Tom’s height?
6.25
Ciid
Wayne is hanging a string of lights 57 feet long around the three sides of his patio, which is adjacent to his house. the length of his patio, the side along the house, is 5 feet longer than twice it's width. Find the length and width of the patio.
Ciid
tyler
tyler
tyler
tyler
SOH = Sine is Opposite over Hypotenuse. CAH= Cosine is Adjacent over Hypotenuse. TOA = Tangent is Opposite over Adjacent.
tyler
H=57 and O=285 figure out what the adjacent?
tyler
Amara currently sells televisions for company A at a salary of $17,000 plus a$100 commission for each television she sells. Company B offers her a position with a salary of $29,000 plus a$20 commission for each television she sells. How many televisions would Amara need to sell for the options to be equal?
what is the quantity and price of the televisions for both options?
karl
Amara has to sell 120 televisions to make 29,000 of the salary of company B. 120 * TV 100= commission 12,000+ 17,000 = of company a salary 29,000 17000+
Ciid
Amara has to sell 120 televisions to make 29,000 of the salary of company B. 120 * TV 100= commission 12,000+ 17,000 = of company a salary 29,000
Ciid
I'm mathematics teacher from highly recognized university.
here a question professor How many soldiers are there in a group of 27 sailors and soldiers if there are four fifths as many sailors as soldiers? can you write out the college you went to with the name of the school you teach at and let me know the answer I've got it to be honest with you
tyler
is anyone else having issues with the links not doing anything?
Yes
Val
chapter 1 foundations 1.2 exercises variables and algebraic symbols
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Joseph would like to make 12 pounds of a coffee blend at a cost of $6.25 per pound. He blends Ground Chicory at$4.40 a pound with Jamaican Blue Mountain at $8.84 per pound. How much of each type of coffee should he use? Samer 4x6.25=$25 coffee blend 4×4.40= $17.60 ground chicory 4x8.84= 35.36 blue mountain. In total they will spend for 12 pounds$77.96 they will spend in total
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DaMarcus and Fabian live 23 miles apart and play soccer at a park between their homes. DaMarcus rode his bike for three-quarters of an hour and Fabian rode his bike for half an hour to get to the park. Fabian’s speed was six miles per hour faster than DaMarcus’ speed. Find the speed of both soccer players.
i need help how to do this is confusing
what kind of math is it?
Danteii
help me to understand
huh, what is the algebra problem
Daniel
How many soldiers are there in a group of 27 sailors and soldiers if there are four fifths many sailors as soldiers?
tyler
What is the domain and range of heaviside
What is the domain and range of Heaviside and signum
Christopher
25-35
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A 50% antifreeze solution is to be mixed with a 90% antifreeze solution to get 200 liters of a 80% solution. How many liters of the 50% solution and how many liters of the 90% solution will be used?
June needs 45 gallons of punch for a party and has 2 different coolers to carry it in. The bigger cooler is 5 times as large as the smaller cooler. How many gallons can each cooler hold?