# 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?

One-fourth of the candies in a bag of M&M’s are red. If there are 23 red candies, how many candies are in the bag?
rectangular field solutions
What is this?
Donna
the proudact of 3x^3-5×^2+3 and 2x^2+5x-4 in z7[x]/ is
?
Choli
a rock is thrown directly upward with an initial velocity of 96feet per second from a cliff 190 feet above a beach. The hight of tha rock above the beach after t second is given by the equation h=_16t^2+96t+190
Usman
Stella bought a dinette set on sale for $725. The original price was$1,299. To the nearest tenth of a percent, what was the rate of discount?
44.19%
Scott
40.22%
Terence
44.2%
Orlando
I don't know
Donna
if you want the discounted price subtract $725 from$1299. then divide the answer by $1299. you get 0.4419... but as percent you get 44.19... but to the nearest tenth... round .19 to .2 and you get 44.2% Orlando you could also just divide$725/$1299 and then subtract it from 1. then you get the same answer. Orlando p mulripied-5 and add 30 to it Tausif Reply p mulripied-5 and add30 Tausif p mulripied-5 and addto30 Tausif Can you explain further Monica Reply p mulripied-5 and add to 30 Tausif How do you find divisible numbers without a calculator? Jacob Reply TAKE OFF THE LAST DIGIT AND MULTIPLY IT 9. SUBTRACT IT THE DIGITS YOU HAVE LEFT. IF THE ANSWER DIVIDES BY 13(OR IS ZERO), THEN YOUR ORIGINAL NUMBER WILL ALSO DIVIDE BY 13!IS DIVISIBLE BY 13 BAINAMA When she graduates college, Linda will owe$43,000 in student loans. The interest rate on the federal loans is 4.5% and the rate on the private bank loans is 2%. The total interest she owes for one year was $1,585. What is the amount of each loan? Ariana Reply Sean took the bus from Seattle to Boise, a distance of 506 miles. If the trip took 7 2/3 hours, what was the speed of the bus? Kirisma Reply 66miles/hour snigdha How did you work it out? Esther s=mi/hr 2/3~0.67 s=506mi/7.67hr = ~66 mi/hr Orlando hello, I have algebra phobia. Subtracting negative numbers always seem to get me confused. Alicia Reply what do you need help in? Felix subtracting a negative....is adding!! Heather look at the numbers if they have different signs, it's like subtracting....but you keep the sign of the largest number... Felix for example.... -19 + 7.... different signs...subtract.... 12 keep the sign of the "largest" number 19 is bigger than 7.... 19 has the negative sign... Therefore, -12 is your answer... Felix —12 Niazmohammad Thanks Felix.l also get confused with signs. Esther Thank you for this Shatey ty Graham think about it like you lost$19 (-19), then found $7(+7). Totally you lost just$12 (-12)
Annushka
I used to struggle a lot with negative numbers and math in general what I typically do is look at it in terms of money I have -$5 in my account I then take out 5 more dollars how much do I have in my account well-$10 ... I also for a long time would draw it out on a number line to visualize it
Meg
practicing with smaller numbers to understand then working with larger numbers helps too and the song/rhyme same sign add and keep opposite signs subtract keep the sign of the bigger # then you'll be exact
Meg
Bruce drives his car for his job. The equation R=0.575m+42 models the relation between the amount in dollars, R, that he is reimbursed and the number of miles, m, he drives in one day. Find the amount Bruce is reimbursed on a day when he drives 220 miles
168.50=R
Heather
john is 5years older than wanjiru.the sum of their years is27years.what is the age of each
46
mustee
j 17 w 11
Joseph
john is 16. wanjiru is 11.
Felix
27-5=22 22÷2=11 11+5=16
Joyce
I don't see where the answers are.
Ed
Cindy and Richard leave their dorm in Charleston at the same time. Cindy rides her bicycle north at a speed of 18 miles per hour. Richard rides his bicycle south at a speed of 14 miles per hour. How long will it take them to be 96 miles apart?
3
Christopher
18t+14t=96 32t=96 32/96 3
Christopher
show that a^n-b^2n is divisible by a-b