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## Algebra 1

### Course: Algebra 1>Unit 5

Lesson 6: Summary: Forms of two-variable linear equations

# Forms of linear equations review

There are three major forms of linear equations: point-slope form, standard form, and slope-intercept form. We review all three in this article.
There are three main forms of linear equations.
Slope-interceptPoint-slopeStandard
y, equals, start color #ed5fa6, m, end color #ed5fa6, x, plus, start color #1fab54, b, end color #1fab54y, minus, start color #7854ab, y, start subscript, 1, end subscript, end color #7854ab, equals, start color #ed5fa6, m, end color #ed5fa6, left parenthesis, x, minus, start color #7854ab, x, start subscript, 1, end subscript, end color #7854ab, right parenthesisA, x, plus, B, y, equals, C
where start color #ed5fa6, m, end color #ed5fa6 is slope and start color #1fab54, b, end color #1fab54 is the y-interceptwhere start color #ed5fa6, m, end color #ed5fa6 is slope and start color #7854ab, left parenthesis, x, start subscript, 1, end subscript, comma, y, start subscript, 1, end subscript, right parenthesis, end color #7854ab is a point on the linewhere A, B, and C are constants

## Example

A line passes through the points left parenthesis, minus, 2, comma, minus, 4, right parenthesis and left parenthesis, minus, 5, comma, 5, right parenthesis. Find the equation of the line in all three forms listed above.
Two of the forms require slope, so let's find that first.
\begin{aligned} \text{slope}=\maroonC m &= \dfrac{\Delta y}{\Delta x}\\\\ &=\dfrac{5-(-4)}{-5-(-2)}\\\\ &=\dfrac{9}{-3} \\\\ &=\maroonC{-3} \end{aligned}
Now we can plug in start color #ed5fa6, m, end color #ed5fa6 and one of the points, say start color #7854ab, left parenthesis, minus, 5, comma, 5, right parenthesis, end color #7854ab, to get point-slope form, y, minus, start color #7854ab, y, start subscript, 1, end subscript, end color #7854ab, equals, start color #ed5fa6, m, end color #ed5fa6, left parenthesis, x, minus, start color #7854ab, x, start subscript, 1, end subscript, end color #7854ab, right parenthesis:
\begin{aligned} y-\purpleD{y_1}&=\maroonC m(x-\purpleD{x_1}) \\\\ y-\purpleD{5}&=\maroonC{-3}(x-\purpleD{(-5)}) \\\\ y-\purpleD{5}&=\maroonC{-3}(x+\purpleD{5}) \end{aligned}
Solving for y, we get slope-intercept form, y, equals, start color #ed5fa6, m, end color #ed5fa6, x, plus, start color #1fab54, b, end color #1fab54:
\begin{aligned} y-{5}&=\maroonC{-3}(x+{5}) \\\\ y-5&=\maroonC{-3}x-15 \\\\ y&=\maroonC{-3}x\greenD{-10} \end{aligned}
And adding 3, x to both sides, we get standard form, A, x, plus, B, y, equals, C:
y, plus, 3, x, equals, minus, 10
Want another example? Check out this video.
Want to practice the different forms yourself? Check out this exercise.
Want a more in-depth review of each form? Check out these review articles: