Linear Equations Worksheet

30 printable problems on slope, writing the equation of a line, converting between slope-intercept and standard form, parallel and perpendicular lines, and solving linear equations, including ones with no solution. Answer key included.

30 problems · answer key included · free, no sign-up

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Teachers and parents: print the student copy for class or homework, or print with the answer key on a separate page. You can also link to this page from your course site or LMS; no account is needed. Problems get harder within each section.

A. Find the slope

Find the slope of the line through each pair of points.

1
$(1, 2)$ and $(4, 8)$
2
$(- 2, 5)$ and $(3, - 5)$
3
$(0, - 3)$ and $(6, 1)$
4
$(- 4, - 1)$ and $(2, - 4)$
5
$(3, 7)$ and $(3, - 2)$
6
$(- 5, 4)$ and $(1, 4)$

B. Write the equation

Write the equation of each line in slope-intercept form, y = mx + b.

7
Slope $3$ , through $(0, - 2)$
8
Slope $- 2$ , through $(1, 5)$
9
Slope $\frac{1}{2}$ , through $(4, 1)$
10
Through $(2, 3)$ and $(6, 11)$
11
Through $(- 3, 4)$ and $(3, - 2)$
12
Through $(- 2, - 5)$ and $(4, - 2)$

C. Convert between forms

Rewrite each equation as directed.

13
Write $3 x + y = 7$ in slope-intercept form.
14
Write $2 x - 5 y = 10$ in slope-intercept form.
15
Write $4 x + 6 y = 12$ in slope-intercept form.
16
Write $y = \frac{3}{4} x - 2$ in standard form $A x + B y = C$ with integer coefficients.
17
Write $y = - 2 x + 5$ in standard form $A x + B y = C$ with integer coefficients.

D. Parallel and perpendicular lines

Parallel lines have equal slopes; perpendicular slopes multiply to −1.

18
Write the equation of the line parallel to $y = 3 x + 1$ through $(2, 4)$ .
19
Write the equation of the line perpendicular to $y = 2 x - 3$ through $(4, 1)$ .
20
Write the equation of the line parallel to $2 x + y = 6$ through $(0, - 1)$ .
21
Are $y = \frac{2}{3} x + 1$ and $3 x + 2 y = 4$ parallel, perpendicular or neither?

E. Linear or not?

Decide whether each equation is linear and explain why.

22
$y = 4 x - 9$
23
$y = x^{2} + 1$
24
$2 x + 3 y = 12$
25
$y = \frac{5}{x}$

F. Solve the equation

Solve. Some equations have no solution or infinitely many.

26
$\frac{x}{2} + 3 = \frac{x}{3} + 5$
27
$0.4 x + 1.2 = 0.1 x + 3$
28
$\frac{2 x - 1}{3} = \frac{x + 4}{2}$
29
$2 (x + 3) = 2 x + 5$
30
$3 (x - 2) + 4 = 3 x - 2$

Answer key

Check your work after you finish each section.

A. Find the slope

1.
$m = 2$
$m = \frac{8 - 2}{4 - 1}$
2.
$m = - 2$
$m = \frac{- 5 - 5}{3 - ( - 2 )}$
3.
$m = \frac{2}{3}$
$m = \frac{1 - ( - 3 )}{6 - 0}$
4.
$m = - \frac{1}{2}$
$m = \frac{- 4 - ( - 1 )}{2 - ( - 4 )}$
5.
$Undefined$
The run is $0$ : a vertical line
6.
$m = 0$
$m = \frac{4 - 4}{1 - ( - 5 )}$ (horizontal line)

B. Write the equation

7.
$y = 3 x - 2$
$b = - 2 - (3) (0) = - 2$
8.
$y = - 2 x + 7$
$b = 5 - (- 2) (1) = 7$
9.
$y = \frac{1}{2} x - 1$
$b = 1 - (\frac{1}{2}) (4) = - 1$
10.
$y = 2 x - 1$
$m = 2$ , then $b = - 1$
11.
$y = - x + 1$
$m = - 1$ , then $b = 1$
12.
$y = \frac{1}{2} x - 4$
$m = \frac{1}{2}$ , then $b = - 4$

C. Convert between forms

13.
$y = - 3 x + 7$
Solve for $y$
14.
$y = \frac{2}{5} x - 2$
Solve for $y$
15.
$y = - \frac{2}{3} x + 2$
Solve for $y$
16.
$3 x - 4 y = 8$
Move the $x$ term to the left, then multiply by $4$
17.
$2 x + y = 5$
Move the $x$ term to the left

D. Parallel and perpendicular lines

18.
$y = 3 x - 2$
Same slope: $m = 3$
19.
$y = - \frac{1}{2} x + 3$
Negative reciprocal slope: $m = - \frac{1}{2}$
20.
$y = - 2 x - 1$
Same slope: $m = - 2$
21.
Perpendicular
Slopes $\frac{2}{3}$ and $- \frac{3}{2}$ multiply to $- 1$

E. Linear or not?

22.
Linear
Written as $y = m x + b$
23.
Not linear
$x$ is squared
24.
Linear
Standard form $A x + B y = C$
25.
Not linear
$x$ is in the denominator

F. Solve the equation

26.
$x = 12$
Multiply every term by $6$ : $3 x + 18 = 2 x + 30$
27.
$x = 6$
$0.3 x = 1.8$
28.
$x = 14$
Cross-multiply: $4 x - 2 = 3 x + 12$
29.
$No solution$
$2 x + 6 = 2 x + 5$ gives $6 = 5$ , which is false
30.
$All real numbers$
Both sides simplify to $3 x - 2$

The forms of a linear equation

Slope-intercept form, $y = m x + b$ : $m$ is the slope and $b$ is the y-intercept. Best for graphing and for reading the slope.

Standard form, $A x + B y = C$ with integers: best for finding intercepts. Convert to slope-intercept form by solving for $y$ .

Point-slope form, $y - y_{1} = m (x - x_{1})$ : best when you know a slope and one point.

Slope from two points: $m = \frac{y _{2} - y _{1}}{x _{2} - x _{1}}$ . Keep the points in the same order on top and bottom.

Parallel and perpendicular: parallel lines share a slope; perpendicular slopes are negative reciprocals, like $2$ and $- \frac{1}{2}$ .

Read the full Linear Equations lesson with worked examples →

Frequently asked questions

What is a linear equation?

An equation whose graph is a straight line. The variables appear only to the first power and are never multiplied together, inside a root or in a denominator. y = 4x − 9 and 2x + 3y = 12 are linear; y = x² + 1 and y = 5/x are not.

How do you find the slope from two points?

Use m = (y₂ − y₁)/(x₂ − x₁): the change in y divided by the change in x. For (1, 2) and (4, 8), m = (8 − 2)/(4 − 1) = 6/3 = 2. If the x-values are equal, the line is vertical and the slope is undefined.

How do you write the equation of a line through two points?

Find the slope m from the two points, substitute one point into y = mx + b and solve for b, then write y = mx + b. For (2, 3) and (6, 11): m = 2, so 3 = 2(2) + b gives b = −1, and the line is y = 2x − 1.

What are the slopes of parallel and perpendicular lines?

Parallel lines have the same slope. Perpendicular lines have slopes that are negative reciprocals, so they multiply to −1: a line perpendicular to y = 2x − 3 has slope −1/2.

Can a linear equation have no solution?

Yes. If solving leads to a false statement, such as 6 = 5 from 2(x + 3) = 2x + 5, there is no solution. If it leads to a true statement, such as 3x − 2 = 3x − 2, every real number is a solution.

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