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### Course: 8th grade (Illustrative Mathematics)>Unit 3

Lesson 4: Lesson 8: Translating to y=mx+b

# Intro to slope-intercept form

Learn about the slope-intercept form of two-variable linear equations, and how to interpret it to find the slope and y-intercept of their line.

#### What you will learn in this lesson

• What is the slope-intercept form of two-variable linear equations
• How to find the slope and the y-intercept of a line from its slope-intercept equation
• How to find the equation of a line given its slope and y-intercept

## What is slope-intercept form?

Slope-intercept is a specific form of linear equations. It has the following general structure. Drum roll ...
y, equals, start color #ed5fa6, m, end color #ed5fa6, x, plus, start color #0d923f, b, end color #0d923f
Here, start color #ed5fa6, m, end color #ed5fa6 and start color #0d923f, b, end color #0d923f can be any two real numbers. For example, these are linear equations in slope-intercept form:
• y, equals, 2, x, plus, 1
• y, equals, minus, 3, x, plus, 2, point, 7
• y, equals, 10, minus, 100, x
On the other hand, these linear equations are not in slope-intercept form:
• 2, x, plus, 3, y, equals, 5
• y, minus, 3, equals, 2, left parenthesis, x, minus, 1, right parenthesis
• x, equals, 4, y, minus, 7
Slope-intercept is the most prominent form of linear equations. Let's dig deeper to learn why this is so.

## The coefficients in slope-intercept form

Besides being neat and simplified, slope-intercept form's advantage is that it gives two main features of the line it represents:
• The slope is start color #ed5fa6, m, end color #ed5fa6.
• The y-coordinate of the y-intercept is start color #0d923f, b, end color #0d923f. In other words, the line's y-intercept is at left parenthesis, 0, comma, start color #0d923f, b, end color #0d923f, right parenthesis.
For example, the line y, equals, start color #ed5fa6, 2, end color #ed5fa6, x, start color #0d923f, plus, 1, end color #0d923f has a slope of start color #ed5fa6, 2, end color #ed5fa6 and a y-intercept at left parenthesis, 0, comma, start color #0d923f, 1, end color #0d923f, right parenthesis:
A coordinate plane. The x- and y-axes each scale by one. The graph of the line is y equals two x plus one. The y-intercept, zero, one, is plotted and labeled. There is a horizontal dotted segment from zero, one to one, one. There is a vertical dotted segment from one, one to one, three. The dotted segments are labeled slope equals two.
The fact that this form gives the slope and the y-intercept is the reason why it is called slope-intercept in the first place!

Problem 1
What is the slope of the line represented by y, equals, 5, x, minus, 7?

Problem 2
What is the slope of the line represented by y, equals, x, plus, 9?

Problem 3
What is the y-intercept of the line represented by y, equals, minus, 6, x, minus, 11?

Problem 4
What is the y-intercept of the line represented by y, equals, 4, x?

Problem 5
What is the slope of the line represented by y, equals, 1, minus, 8, x?

Problem 6
Which lines have a y-intercept at left parenthesis, 0, comma, 4, right parenthesis?

Reflection question
How do we find the slope of a line that is given in slope-intercept form?

Challenge problem 1
A coordinate plane. The horizontal axis is labeled x. The vertical axis is labeled y. The graph of the line is falling from left to right and has a positive y-intercept.
Which of these can be the equation of the line?

Challenge problem 2
Write an equation of a line whose slope is 10 and y-intercept is left parenthesis, 0, comma, minus, 20, right parenthesis.

## Why does this work?

You might be wondering how it is that in slope-intercept form, start color #ed5fa6, m, end color #ed5fa6 gives the slope and start color #0d923f, b, end color #0d923f gives the y-intercept.
Can this be some sort of magic? Well, it certainly is not magic. In math, there's always a justification. In this section we'll take a look at this property using the equation y, equals, start color #ed5fa6, 2, end color #ed5fa6, x, plus, start color #0d923f, 1, end color #0d923f as an example.

### Why $\greenE{b}$start color #0d923f, b, end color #0d923f gives the $y$y-intercept

At the y-intercept, the x-value is always zero. So if we want to find the y-intercept of y, equals, start color #ed5fa6, 2, end color #ed5fa6, x, plus, start color #0d923f, 1, end color #0d923f, we should substitute x, equals, 0 and solve for y.
\begin{aligned} y&=\maroonC{2}x+\greenE{1} \\\\ &=\maroonC{2}\cdot 0+\greenE{1}&\gray{\text{Substitute }x=0} \\\\ &=0+\greenE{1} \\\\ &=\greenE{1} \end{aligned}
We see that at the y-intercept, start color #ed5fa6, 2, end color #ed5fa6, x becomes zero, and therefore we are left with y, equals, start color #0d923f, 1, end color #0d923f.

### Why $\maroonC{m}$start color #ed5fa6, m, end color #ed5fa6 gives the slope

Let's refresh our memories about what slope is exactly. Slope is the ratio of the change in y over the change in x between any two points on the line.
start text, S, l, o, p, e, end text, equals, start fraction, start text, C, h, a, n, g, e, space, i, n, space, end text, y, divided by, start text, C, h, a, n, g, e, space, i, n, space, end text, x, end fraction
If we take two points where the change in x is exactly 1 unit, then the change in y will be equal to the slope itself.
start text, S, l, o, p, e, end text, equals, start fraction, start text, C, h, a, n, g, e, space, i, n, space, end text, y, divided by, 1, end fraction, equals, start text, C, h, a, n, g, e, space, i, n, space, end text, y
Now let's look at what happens to the y-values in the equation y, equals, start color #ed5fa6, 2, end color #ed5fa6, x, plus, start color #0d923f, 1, end color #0d923f as the x-values constantly increase by 1 unit.
xy
0start color #0d923f, 1, end color #0d923f, plus, 0, dot, start color #ed5fa6, 2, end color #ed5fa6equals, start color #0d923f, 1, end color #0d923f
1start color #0d923f, 1, end color #0d923f, plus, 1, dot, start color #ed5fa6, 2, end color #ed5fa6equals, start color #0d923f, 1, end color #0d923f, plus, start color #ed5fa6, 2, end color #ed5fa6
2start color #0d923f, 1, end color #0d923f, plus, 2, dot, start color #ed5fa6, 2, end color #ed5fa6equals, start color #0d923f, 1, end color #0d923f, plus, start color #ed5fa6, 2, end color #ed5fa6, plus, start color #ed5fa6, 2, end color #ed5fa6
3start color #0d923f, 1, end color #0d923f, plus, 3, dot, start color #ed5fa6, 2, end color #ed5fa6equals, start color #0d923f, 1, end color #0d923f, plus, start color #ed5fa6, 2, end color #ed5fa6, plus, start color #ed5fa6, 2, end color #ed5fa6, plus, start color #ed5fa6, 2, end color #ed5fa6
4start color #0d923f, 1, end color #0d923f, plus, 4, dot, start color #ed5fa6, 2, end color #ed5fa6equals, start color #0d923f, 1, end color #0d923f, plus, start color #ed5fa6, 2, end color #ed5fa6, plus, start color #ed5fa6, 2, end color #ed5fa6, plus, start color #ed5fa6, 2, end color #ed5fa6, plus, start color #ed5fa6, 2, end color #ed5fa6
We see that each time x increases by 1 unit, y increases by start color #ed5fa6, 2, end color #ed5fa6 units. This is because x determines the multiple of start color #ed5fa6, 2, end color #ed5fa6 in the calculation of y.
As stated above, the change in y that corresponds to x increasing by 1 unit is equal to the slope of the line. For this reason, the slope is start color #ed5fa6, 2, end color #ed5fa6.
Challenge problem 3
A coordinate plane. The x- and y-axes each scale by one. The graph of a line goes through the points zero, negative three and one, one. Both of these points are plotted and labeled.
Complete the equation of the line.
y, equals