# 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.

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at least 20
Ayla
what are the steps?
Alicia
6.4 jobs
Grahame
32
Grahame
1600+2400= total amount with expenses. 4000/125= number of jobs needed to make that min profit of 2400. answer is 32
Orlando
He must work 32 jobs to make a profit
POP
what is algebra
repeated addition and subtraction of the order of operations. i love algebra I'm obsessed.
Shemiah
hi
Krekar
Eric here. I'm a parent. 53 years old. I have never taken algebra. I want to learn.
Eric
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?
they are 92 candies in the bag
POP
rectangular field solutions
What is this?
Donna
t
muqtaar
the proudact of 3x^3-5×^2+3 and 2x^2+5x-4 in z7[x]/ is
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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
Tausif
how
muqtaar
Can you explain further
p mulripied-5 and add to 30
Tausif
-5p+30?
Corey
p=-5+30
Jacob