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## Algebra basics (ASL)

### Course: Algebra basics (ASL)>Unit 4

Lesson 7: Writing slope-intercept equations (ASL)

# Writing slope-intercept equations (ASL)

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.

## Check your understanding

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.

## Check your understanding

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.