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

Lesson 8: Extra practice: Slope

# Writing slope-intercept equations

Learn how to find the slope-intercept equation of a line from two points on that line.
If you haven't read it yet, you might want to start with our introduction to slope-intercept form.

## Writing equations from $y$y-intercept and another point

A first quadrant coordinate plane. The x- and y-axes each scale by one. A graph of a line goes through the points zero, three and two, seven, which are plotted and labeled.
Let's write the equation of the line that passes through the points left parenthesis, 0, comma, 3, right parenthesis and left parenthesis, 2, comma, 7, right parenthesis in slope-intercept form.
Recall that in the general slope-intercept equation y, equals, start color #ed5fa6, m, end color #ed5fa6, x, plus, start color #0d923f, b, end color #0d923f, the slope is given by start color #ed5fa6, m, end color #ed5fa6 and the y-intercept is given by start color #0d923f, b, end color #0d923f.

### Finding $\greenE b$start color #0d923f, b, end color #0d923f

The y-intercept of the line is left parenthesis, 0, comma, start color #0d923f, 3, end color #0d923f, right parenthesis, so we know that start color #0d923f, b, end color #0d923f, equals, start color #0d923f, 3, end color #0d923f.

### Finding $\maroonC m$start color #ed5fa6, m, end color #ed5fa6

Recall that the slope of a line 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
Therefore, this is the slope between the points left parenthesis, 0, comma, 3, right parenthesis and left parenthesis, 2, comma, 7, right parenthesis:
\begin{aligned}\maroonC{m}&=\dfrac{\text{Change in }y}{\text{Change in }x} \\\\ &=\dfrac{7-3}{2-0} \\\\ &=\dfrac{4}{2} \\\\ &=\maroonC{2}\end{aligned}
In conclusion, the equation of the line is y, equals, start color #ed5fa6, 2, end color #ed5fa6, x, start color #0d923f, plus, 3, end color #0d923f.

Problem 1
A first quadrant coordinate plane. The x- and y-axes each scale by one. A graph of a line goes through the points zero, five and four, nine, which are plotted and labeled.
Write the equation of the line.

Problem 2
A first quadrant coordinate plane. The x- and y-axes each scale by one. A graph of a line goes through the points zero, eight and three, two which are plotted and labeled.
Write the equation of the line.

## Writing equations from any two points

A first quadrant coordinate plane. The x- and y-axes each scale by one. A graph of a line goes through the points two, five and four, nine, which are plotted and labeled.
Let's write the equation of the line that passes through left parenthesis, 2, comma, 5, right parenthesis and left parenthesis, 4, comma, 9, right parenthesis in slope-intercept form.
Note that we are not given the y-intercept of the line. This makes things a little bit more difficult, but we are not afraid of a challenge!

### Finding $\maroonC m$start color #ed5fa6, m, end color #ed5fa6

\begin{aligned} \maroonC{m}&=\dfrac{\text{Change in }y}{\text{Change in }x} \\\\ &=\dfrac{9-5}{4-2} \\\\ &=\dfrac{4}{2} \\\\ &=\maroonC{2} \end{aligned}

### Finding $\greenE b$start color #0d923f, b, end color #0d923f

We know that the line is of the form y, equals, start color #ed5fa6, 2, end color #ed5fa6, x, plus, start color #0d923f, b, end color #0d923f, but we still need to find start color #0d923f, b, end color #0d923f. To do that, we substitute the point left parenthesis, 2, comma, 5, right parenthesis into the equation.
Because any point on a line must satisfy that line’s equation, we get an equation that we can solve to find start color #0d923f, b, end color #0d923f.
\begin{aligned}y&=\maroonC{2}\cdot x+\greenE{b}\\\\ 5&=\maroonC{2}\cdot 2+\greenE{b}&\gray{x=2\text{ and }y=5}\\\\ 5&=4+\greenE{b}\\\\ \greenE{1}&=\greenE{b} \end{aligned}
In conclusion, the equation of the line is y, equals, start color #ed5fa6, 2, end color #ed5fa6, x, start color #0d923f, plus, 1, end color #0d923f.

Problem 3
A first quadrant coordinate plane. The x- and y-axes each scale by one. A graph of a line goes through the points one, four and three, ten, which are plotted and labeled.
Write the equation of the line.

Problem 4
A first quadrant coordinate plane. The x- and y-axes each scale by one. A graph of a line goes through the points two, nine and four, one, which are plotted and labeled.
Write the equation of the line.

Challenge problem
A line passes through the points left parenthesis, 5, comma, 35, right parenthesis and left parenthesis, 9, comma, 55, right parenthesis.
Write the equation of the line.

## Want to join the conversation?

• I think I may need to give up and be a farmer because this is to hard
• Alexis, I am in my 40's relearning this. Don't give up. Plus, farmers use TONs (TONNES, depending on where you live) of maths. Keep practicing, every wrong answer is still learning.
• everything about what we are learning I don't understand
• You may need to go back to wherever you feel comfortable with, and learn upward again slowly. You can do this within Khan Academy.
• i think i'll just sell corn on the street
• fr but i think lemonade would be better
• What is the rule with deciding which point value gets subtracted from the other?
• It doesn't matter. The only difference is that there's a sign change, but since this happens both for 𝛥𝑦 as for 𝛥𝑥 these changes cancel out when we divide the two (𝛥𝑦∕𝛥𝑥).

𝛥𝑦 = 𝑦₂ − 𝑦₁ = −(𝑦₁ − 𝑦₂)
𝛥𝑥 = 𝑥₂ − 𝑥₁ = −(𝑥₁ − 𝑥₂)

𝛥𝑦∕𝛥𝑥 = (𝑦₂ − 𝑦₁)∕(𝑥₂ − 𝑥₁) = −(𝑦₁ − 𝑦₂)∕(−(𝑥₁ − 𝑥₂)) = (𝑦₁ − 𝑦₂)∕(𝑥₁ − 𝑥₂)
• On number 4, why would b=17?
• This is the process they show you:
y =−4x+b
9 =−4⋅2+b
9 =−8+b
17=b
----------
In THIS ("9 =−8+b") step they are subtracting -8 from 9 which looks like this: 9-(-8). We know that two negative operations make a positive sum, which is why the answer is 17 : 9-(-8) = 17.

• Bruh this is hard to do. I still dont undertand
• no way that this makes a single drop of sense
• bro why does hurt my brain
like i did fine on my Quiz
• how do you change 7x+3y=3 into slope intercept form
• For something to be in slope-intercept for, y needs to be isolated on one side of the equation.
Here's how you'd do that:
7x + 3y = 3

Subtract 7x from both sides:
7x + 3y - 7x = 3 - 7x

Simplify:
3y = 3 - 7x

Now, divide both sides by 3:
3y/3 = 3/3 - 7x/3

Simplify:
y = 1 - (7/3)x

Usually, we have the intercept (in this case 1) on the right side, so simply move it:
y = -(7/3)x + 1