# 4.5 Use the slope–intercept form of an equation of a line

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By the end of this section, you will be able to:
• Recognize the relation between the graph and the slope–intercept form of an equation of a line
• Identify the slope and y-intercept form of an equation of a line
• Graph a line using its slope and intercept
• Choose the most convenient method to graph a line
• Graph and interpret applications of slope–intercept
• Use slopes to identify parallel lines
• Use slopes to identify perpendicular lines

Before you get started, take this readiness quiz.

1. Add: $\frac{x}{4}+\frac{1}{4}.$
If you missed this problem, review [link] .
2. Find the reciprocal of $\frac{3}{7}.$
If you missed this problem, review [link] .
3. Solve $2x-3y=12\phantom{\rule{0.2em}{0ex}}\text{for}\phantom{\rule{0.2em}{0ex}}y$ .
If you missed this problem, review [link] .

## Recognize the relation between the graph and the slope–intercept form of an equation of a line

We have graphed linear equations by plotting points, using intercepts, recognizing horizontal and vertical lines, and using the point–slope method. Once we see how an equation in slope–intercept form and its graph are related, we’ll have one more method we can use to graph lines.

In Graph Linear Equations in Two Variables , we graphed the line of the equation $y=\frac{1}{2}x+3$ by plotting points. See [link] . Let’s find the slope of this line.

The red lines show us the rise is 1 and the run is 2. Substituting into the slope formula:

$\begin{array}{ccc}\hfill m& =\hfill & \frac{\text{rise}}{\text{run}}\hfill \\ \hfill m& =\hfill & \frac{1}{2}\hfill \end{array}$

What is the y -intercept of the line? The y -intercept is where the line crosses the y -axis, so y -intercept is $\left(0,3\right)$ . The equation of this line is:

Notice, the line has:

When a linear equation is solved for $y$ , the coefficient of the $x$ term is the slope and the constant term is the y -coordinate of the y -intercept. We say that the equation $y=\frac{1}{2}x+3$ is in slope–intercept form.

## Slope-intercept form of an equation of a line

The slope–intercept form of an equation of a line with slope $m$ and y -intercept, $\left(0,b\right)$ is,

$y=mx+b$

Sometimes the slope–intercept form is called the “ y -form.”

Use the graph to find the slope and y -intercept of the line, $y=2x+1$ .

Compare these values to the equation $y=mx+b$ .

## Solution

To find the slope of the line, we need to choose two points on the line. We’ll use the points $\left(0,1\right)$ and $\left(1,3\right)$ .

 Find the rise and run. Find the y -intercept of the line. The y -intercept is the point (0, 1).

The slope is the same as the coefficient of $x$ and the y -coordinate of the y -intercept is the same as the constant term.

Use the graph to find the slope and y -intercept of the line $y=\frac{2}{3}x-1$ . Compare these values to the equation $y=mx+b$ .

slope $m=\frac{2}{3}$ and y -intercept $\left(0,-1\right)$

Use the graph to find the slope and y -intercept of the line $y=\frac{1}{2}x+3$ . Compare these values to the equation $y=mx+b$ .

slope $m=\frac{1}{2}$ and y -intercept $\left(0,3\right)$

## Identify the slope and y -intercept from an equation of a line

In Understand Slope of a Line , we graphed a line using the slope and a point. When we are given an equation in slope–intercept form, we can use the y -intercept as the point, and then count out the slope from there. Let’s practice finding the values of the slope and y -intercept from the equation of a line.

Identify the slope and y -intercept of the line with equation $y=-3x+5$ .

## Solution

We compare our equation to the slope–intercept form of the equation.

 Write the equation of the line. Identify the slope. Identify the y -intercept.

4x+7y=29,x+3y=11 substitute method of linear equation
substitute method of linear equation
Srinu
Solve one equation for one variable. Using the 2nd equation, x=11-3y. Substitute that for x in first equation. this will find y. then use the value for y to find the value for x.
bruce
I want to learn
Elizebeth
help
Elizebeth
I want to learn. Please teach me?
Wayne
1) Use any equation, and solve for any of the variables. Since the coefficient of x (the number in front of the x) in the second equation is 1 (it actually isn't shown, but 1 * x = x), use that equation. Subtract 3y from both sides (this isolates the x on the left side of the equal sign).
bruce
2) This results in x=11-3y. x is note in terms of y. Use that as the value of x and substitute for all x in the first equation. The first equation becomes 4(11-3y)+7y =29. Note that the only variable left in the first equation is the y. If you have multiple variable, then something is wrong.
bruce
3) Distribute (multiply) the 4 across 11-3y to get 44-12y. Add this to the 7y. So, the equation is now 44-5y=29.
bruce
4) Solve 44-5y=29 for y. Isolate the y by subtracting 44 from birth sides, resulting in -5y=-15. Now, divide birth sides by -5 (since you have -5y). This results in y=3. You now have the value of one variable.
bruce
5) The last step is to take the value of y from Step 4) and substitute into the 2nd equation. Therefore: x+3y=11 becomes x+3(3)=11. Then multiplying, x+9=11. Finally, solve for x by subtracting 9 from both sides. Therefore, x=2.
bruce
6) The ordered pair of (2, 3) is the proposed solution. To check, substitute those values into either equation. If the result is true, then the solution is correct. 4(2)+7(3)=8+21=29. TRUE! Finished.
bruce
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Two sisters like to compete on their bike rides. Tamara can go 4 mph faster than her sister, Samantha. If it takes Samantha 1 hours longer than Tamara to go 80 miles, how fast can Samantha ride her bike? Got questions? Get instant answers now!
how do u solve that question
Seera
Two sisters like to compete on their bike rides. Tamara can go 4 mph faster than her sister, Samantha. If it takes Samantha 1 hours longer than Tamara to go 80 miles, how fast can Samantha ride her bike?
Seera
Speed=distance ÷ time
Tremayne
x-3y =1; 3x-2y+4=0 graph
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June needs 48 gallons of punch for a party and has two different coolers to carry it in. The bigger cooler is five times as large as the smaller cooler. How many gallons can each cooler hold?
Susanna if the first cooler holds five times the gallons then the other cooler. The big cooler holda 40 gallons and the 2nd will hold 8 gallons is that correct?
Georgie
@Susanna that person is correct if you divide 40 by 8 you can see it's 5 it's simple
Ashley
@Geogie my bad that was meant for u
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Hi everyone, I'm glad to be connected with you all. from France.
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