Linear Equations & Slope Cheat Sheet

A printable reference covering slope, slope-intercept form, standard form, point-slope form, and linear graphs for grades 7-9.

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Linear equations describe relationships that change at a constant rate. This cheat sheet helps students connect tables, graphs, equations, and real-world situations. It is useful for quickly identifying slope, intercepts, and the correct equation form. Students in grades 7-9 use these skills often in algebra, coordinate graphing, and word problems. The most important idea is that slope measures the rate of change between two points. The slope formula is $m=\frac{y_2-y_1}{x_2-x_1}$ , and slope-intercept form is $y=mx+b$ . Point-slope form, $y-y_1=m(x-x_1)$ , is useful when a point and slope are known. Standard form, $Ax+By=C$ , helps with intercepts and comparing linear equations.

Key Facts

The slope between two points is $m=\frac{y_2-y_1}{x_2-x_1}$ , where the numerator is the vertical change and the denominator is the horizontal change.
Slope-intercept form is $y=mx+b$ , where $m$ is the slope and $b$ is the $y$ -intercept.
Point-slope form is $y-y_1=m(x-x_1)$ , which uses a known point $(x_1,y_1)$ and slope $m$ .
Standard form is $Ax+By=C$ , where $A$ , $B$ , and $C$ are constants and $A$ is usually nonnegative.
A horizontal line has slope $m=0$ and an equation like $y=c$ .
A vertical line has undefined slope and an equation like $x=c$ .
Parallel lines have equal slopes, so if two nonvertical lines are parallel, then $m_1=m_2$ .
Perpendicular lines have slopes whose product is $-1$ , so $m_1m_2=-1$ when neither line is vertical or horizontal.

Vocabulary

Slope: Slope is the rate of change of a line, calculated by $m=\frac{\text{rise}}{\text{run}}$ .
Y-intercept: The $y$ -intercept is the point where a graph crosses the $y$ -axis, usually written as $(0,b)$ .
X-intercept: The $x$ -intercept is the point where a graph crosses the $x$ -axis, found by setting $y=0$ .
Linear equation: A linear equation is an equation whose graph is a straight line and has a constant rate of change.
Slope-intercept form: Slope-intercept form is $y=mx+b$ , where the slope and $y$ -intercept can be read directly.
Point-slope form: Point-slope form is $y-y_1=m(x-x_1)$ and is used when one point and the slope are known.

Common Mistakes to Avoid

Using $\frac{x_2-x_1}{y_2-y_1}$ for slope: This reverses rise and run, so the slope becomes the reciprocal of the correct value.
Mixing the order of points in the slope formula: If you start with $y_2-y_1$ in the numerator, you must use $x_2-x_1$ in the denominator.
Confusing the $y$ -intercept with the slope in $y=mx+b$ : The coefficient $m$ is the slope, while $b$ is the value of $y$ when $x=0$ .
Treating vertical lines as having slope $0$ : A vertical line has undefined slope because its run is $0$ , which would require division by $0$ .
Forgetting to distribute in point-slope form: In $y-y_1=m(x-x_1)$ , the slope $m$ must multiply both terms inside the parentheses.

Practice Questions

1 Find the slope of the line through $(2,5)$ and $(6,13)$ .
2 Write the equation of the line with slope $m=-3$ and $y$ -intercept $b=4$ in slope-intercept form.
3 A line passes through $(1,-2)$ with slope $m=5$ . Write its equation in point-slope form and then slope-intercept form.
4 Explain how you can tell from a graph whether a line has positive slope, negative slope, zero slope, or undefined slope.