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## 8th grade (Illustrative Mathematics)

### 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!

## Check your understanding

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?
Choose 1 answer:

Problem 4
What is the y-intercept of the line represented by y, equals, 4, x?
Choose 1 answer:

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?
Choose all answers that apply:

Reflection question
How do we find the slope of a line that is given in slope-intercept form?
Choose 1 answer:

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?
Choose 1 answer:

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

## Want to join the conversation?

• What if m=0?
(41 votes)
• If the slope is 0, is a horizontal line. It makes sense if you think about it. Each time we increase one x, increase y by 0.
(105 votes)
• how do you find the slope and intercept on a graph?
(28 votes)
• To find the y-intercept, find where the line hits the y-axis. To find the x-intercept (which wasn't mentioned in the text), find where the line hits the x-axis. To find the slope, find two points on the line then do y2-y1/x2-x1 the numbers are subscripts.
Hope that helped.
(35 votes)
• I dont understand this whole thing at all PLEASE HELP!
(17 votes)
• The slope-intercept form of a linear equation is where one side contains just "y". So, it will look like: y = mx + b where "m" and "b" are numbers.
This form of the equation is very useful. The coefficient of "x" (the "m" value) is the slope of the line. And, the constant (the "b" value) is the y-intercept at (0, b)
So, if you are given an equation like: y = 2/3 (x) -5
We can tell that the slope of the line = 2/3 and the y-intercept is at (0, -5)
Hope this helps.
(23 votes)
• how does an equation result to an answer?
(22 votes)
• The equation results in how to graph the line on a graph. If they give you the x value then you would plug that in and it would tell you the answer in y.
(8 votes)
• Why should I learn this and what can I use this for in the future.
(9 votes)
• slopes are all over the place in the real world, so it depends on what you plan to do in life of how much you use this. Art, building, science, engineering, finance, statistics, etc. all use linear functions.
(10 votes)
• Is it ever possible that the slope of a linear function can fluctuate? Or is the slope always a fixed value?
(10 votes)
• It is a fixed value, but it could possibly look different. So if the slope is 2, you might find points that create a slope of 4/2 or 6/3 or 8/4 or maybe even 1/.5, but each of these will reduce to the same slope of 2.
(6 votes)
• say you have a problem like (3,1) slope= 4/3. how would you work that out
(3 votes)
• Pretty late here, but for anyone else reading, I'll assume they meant how you find the slope intercept using only these values.

Since we know the slope is 4/3, we can conclude that: y = 4/3 * x ... But what is the constant, the y axis intercept point?

You can solve for it by doing: 1 = 4/3 * 3 + c... We know the values for x and y at some point in the line, but we want to know the constant, c. You can solve this algebraically.
1 = 4/3 * 3 + c
1 = 4 + c
1 - 4 = 4 - 4 + c
-3 = c

The slope intercept equation is: y = 4/3 * x - 3

The y axis intercept point is: (0 , -3)

I just started learning this so if anyone happens across this and spots an error lemme know.
(7 votes)
• How do I find the x intercept?
(3 votes)
• You use y=0 in the equation and calculate "x"
(6 votes)
• I am so lost I need help :((
(5 votes)
• The start of the lesson states what you should have some understanding of, so the first question is do you have some understanding of these two concepts?
"You should know what two-variable linear equations are. Specifically, you should know that the graph of such equations is a line. If this is new to you, check out our intro to two-variable equations.
You should also be familiar with the following properties of linear equations: y-intercept and x-intercept and slope."
If these are an issue, you need to go back and review these concepts. If you understand these, then you need to be more specific on where you are struggling. This is just an intro, so it is basically identifying slope and intercept from an equation. Which checks do not make sense?
(2 votes)
• What is the purpose of articles? I can never understand them. Videos are much easier to understand. Why can't they just put the complicated things in a video?
(4 votes)