# 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?

A veterinarian is enclosing a rectangular outdoor running area against his building for the dogs he cares for. He needs to maximize the area using 100 feet of fencing. The quadratic equation A=x(100−2x) gives the area, A , of the dog run for the length, x , of the building that will border the dog run. Find the length of the building that should border the dog run to give the maximum area, and then find the maximum area of the dog run.
ggfcc
Mike
Washing his dad’s car alone, eight year old Levi takes 2.5 hours. If his dad helps him, then it takes 1 hour. How long does it take the Levi’s dad to wash the car by himself?
1,75hrs
Mike
I'm going to guess. Divide Levi's time by 2. Then divide 1 hour by 2. 1.25 + 0.5 = 1.3?
John
Oops I mean 1.75
John
I'm guessing this because since I have divide 1 hour by 2, I have to do the same for the 2.5 hours it takes Levi by himself.
John
Drew burned 1,800 calories Friday playing 1 hour of basketball and canoeing for 2 hours. On Saturday, he spent 2 hours playing basketball and 3 hours canoeing and burned 3,200 calories. How many calories did he burn per hour when playing basketball?
Brandon has a cup of quarters and dimes with a total value of $3.80. The number of quarters is four less than twice the number of dimes. How many quarters and how many dimes does Brandon have? Kendra Reply Tickets to a Broadway show cost$35 for adults and $15 for children. The total receipts for 1650 tickets at one performance were$47,150. How many adult and how many child tickets were sold?
825
Carol
Arnold invested $64,000, some at 5.5% interest and the rest at 9%. How much did he invest at each rate if he received$4,500 in interest in one year? How do I do this
how to square
easiest way to find the square root of a large number?
Jackie
the accompanying figure shows known flow rates of hydrocarbons into and out of a network of pipes at an oil refinery set up a linear system whose solution provides the unknown flow rates (b) solve the system for the unknown flow rates (c) find the flow rates and directions of flow if x4=50and x6=0
What is observation
I'm confused by the question. Can you describe or explain the math question it pertains to?
Melissa
there is no math to it because all you use is your vision or gaze to the sorrounding areas
Cesarp
Teegan likes to play golf. He has budgeted $60 next month for the driving range. It costs him$10.55 for a bucket of balls each time he goes. What is the maximum number of times he can go to the driving range next month?
5 times max
Anton
Felecia left her home to visit her daughter, driving 45mph. Her husband waited for the dog sitter to arrive and left home 20 minutes, or 1/3 hour later. He drove 55mph to catch up to Felecia. How long before he reaches her?
35 min
Debra
Carmen wants to tile the floor of his house. He will need 1,000 square feet of tile. He will do most of the floor with a tile that costs $1.50 per square foot, but also wants to use an accent tile that costs$9.00 per square foot. How many square feet of each tile should he plan to use if he wants the overall cost to be $3 per square foot? Parker Reply what you wanna get Cesar 800 sq. ft @$1.50 & 200 sq. ft @ $9.00 Marco Geneva treated her parents to dinner at their favorite restaurant. The bill was$74.25. Geneva wants to leave 16 % of the total - bill as a tip. How much should the tip be?
74.25 × .16 then get the total and that will be your tip
David
$74.25 x 0.16 =$11.88 total bill: $74.25 +$11.88 = $86.13 ericka yes and tip 16% will be$11.88
David
what is the shorter way to do it
Priam has dimes and pennies in a cup holder in his car. The total value of the coins is \$4.21. The number of dimes is three less than four times the number of pennies. How many dimes and how many pennies are in the cup?