Slope Forms (Slope-Intercept, Point-Slope, Standard) Cheat Sheet

This cheat sheet covers the three main forms of linear equations: slope-intercept form, point-slope form, and standard form. Students need these forms to graph lines, write equations from given information, and compare linear relationships. Each form shows different information quickly, so choosing the right one can make a problem much easier.

The sheet is designed as a clear reference for classwork, homework, and test review.

Slope-intercept form, $y = mx + b$ , shows the slope and $y$ -intercept directly. Point-slope form, $y - y_1 = m(x - x_1)$ , is useful when you know a point and the slope. Standard form, $Ax + By = C$ , is useful for finding intercepts and comparing equations.

Converting between forms helps students graph accurately and understand that the same line can be written in different ways.

Key Facts

Slope-intercept form is $y = mx + b$ , where $m$ is the slope and $b$ is the $y$ -intercept.
Point-slope form is $y - y_1 = m(x - x_1)$ , where $(x_1,y_1)$ is a point on the line and $m$ is the slope.
Standard form is $Ax + By = C$ , where $A$ , $B$ , and $C$ are usually integers and $A$ is often written as nonnegative.
Slope is calculated by $m = \frac{y_2 - y_1}{x_2 - x_1}$ when two points $(x_1,y_1)$ and $(x_2,y_2)$ are known.
To graph $y = mx + b$ , start at the $y$ -intercept $(0,b)$ and use the slope $m = \frac{\text{rise}}{\text{run}}$ .
To find the $x$ -intercept from $Ax + By = C$ , set $y = 0$ and solve $Ax = C$ .
To find the $y$ -intercept from $Ax + By = C$ , set $x = 0$ and solve $By = C$ .
Horizontal lines have slope $m = 0$ and equations like $y = c$ , while vertical lines have undefined slope and equations like $x = c$ .

Vocabulary

Slope: Slope is the rate of change of a line, found by comparing vertical change to horizontal change.
Y-intercept: The $y$ -intercept is the point where a line crosses the $y$ -axis, written as $(0,b)$ .
X-intercept: The $x$ -intercept is the point where a line crosses the $x$ -axis, found by setting $y = 0$ .
Slope-intercept form: Slope-intercept form is $y = mx + b$ , which directly shows the slope and the $y$ -intercept.
Point-slope form: Point-slope form is $y - y_1 = m(x - x_1)$ , which uses one point on a line and the slope.
Standard form: Standard form is $Ax + By = C$ , a linear equation form often used to find intercepts.

Common Mistakes to Avoid

Confusing slope and $y$ -intercept in $y = mx + b$ is wrong because $m$ gives the rate of change and $b$ gives the starting point on the $y$ -axis.
Using $m = \frac{x_2 - x_1}{y_2 - y_1}$ is wrong because slope must be vertical change over horizontal change, $m = \frac{y_2 - y_1}{x_2 - x_1}$ .
Forgetting the subtraction signs in $y - y_1 = m(x - x_1)$ is wrong because the signs depend on the coordinates of the known point.
Treating $Ax + By = C$ as slope-intercept form is wrong because the slope and $y$ -intercept are not visible until the equation is solved for $y$ .
Calling a vertical line's slope $0$ is wrong because horizontal lines have slope $0$ , while vertical lines have undefined slope.

Practice Questions

1 Write the equation of a line in slope-intercept form with slope $m = 3$ and $y$ -intercept $b = -4$ .
2 Find the slope of the line through $(2,5)$ and $(6,13)$ using $m = \frac{y_2 - y_1}{x_2 - x_1}$ .
3 Convert $2x + 3y = 12$ to slope-intercept form and identify the slope and $y$ -intercept.
4 Explain which form of a linear equation is most useful when you know a point on the line and the slope, and explain why.

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Slope Forms (Slope-Intercept, Point-Slope, Standard) Cheat Sheet

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