5.4 Solving Linear Equations with a General Strategy

Solving Linear Equations with a General Strategy

A source-bound narrated opening with a worked example, an eight-second thinking pause, a checked result, English captions, and a transcript.

Read the transcript Download the video Narration credits and licences

Solve Equations with Variables and Constants on Both Sides

In the next example, the equation has variables and constants on both sides. We’ll collect the variable terms to one side and the constants to the other side.

Example 1

Solve: $7 x + 5 = 6 x + 2 .$

Show solution

Start by choosing which side will be the variable side and which side will be the constant side. The variable terms are $7 x$ and $6 x.$ Since $7$ is greater than $6,$ make the left side the variable side and so the right side will be the constant side.


Collect the variable terms to the left side by subtracting $6 x$ from both sides.
Simplify.
Now, collect the constants to the right side by subtracting 5 from both sides.
Simplify.
The solution is $x equals −3$ .
Check: Let $x equals −3$ .

We’ll summarize the steps we took so you can easily refer to them.

It is a good idea to make the variable side the one in which the variable has the larger coefficient. This usually makes the arithmetic easier.

Example 2

Solve: $6 n - 2 = −3 n + 7 .$

Show solution

We have $6 n$ on the left and $−3 n$ on the right. Since $6 greater than minus 3,$ make the left side the “variable” side.


We don't want variables on the right side—add $3 n$ to both sides to leave only constants on the right.
Combine like terms.
We don't want any constants on the left side, so add 2 to both sides.
Simplify.
The variable term is on the left and the constant term is on the right. To get the coefficient of $n$ to be one, divide both sides by 9.
Simplify.
Check: Substitute 1 for $n$ .

Example 3

Solve: $2 a - 7 = 5 a + 8 .$

Show solution

This equation has $2 a$ on the left and $5 a$ on the right. Since $5 greater than 2,$ make the right side the variable side and the left side the constant side.


Subtract $2 a$ from both sides to remove the variable term from the left.
Combine like terms.
Subtract 8 from both sides to remove the constant from the right.
Simplify.
Divide both sides by 3 to make 1 the coefficient of $a$ .	$A mathematical equation is shown with fractions. On the left, -15 is divided by 3, with the 3 in red. On the right, 3a is divided by 3, with the 3 in red. An equals sign separates the two fractions.$
Simplify.
Check: Let $a equals −5$ .

Note that we could have made the left side the variable side instead of the right side, but it would have led to a negative coefficient on the variable term. While we could work with the negative, there is less chance of error when working with positives. The strategy outlined above helps avoid the negatives!

To solve an equation with fractions, we still follow the same steps to get the solution.

Example 4

Solve: $\frac{3}{2} x + 5 = \frac{1}{2} x - 3 .$

Show solution

Since $\frac{3}{2} > \frac{1}{2},$ make the left side the variable side and the right side the constant side.


Subtract $\frac{1}{2} x$ from both sides.	$A mathematical equation displayed as 3/2x - 1/2x + 5 = 1/2x - 1/2x - 3, featuring fractional coefficients, the variable 'x', and integer constants, with some terms highlighted in red.$
Combine like terms.
Subtract 5 from both sides.
Simplify.
Check: Let $x equals −8$ .

We follow the same steps when the equation has decimals, too.

Example 5

Solve: $3.4 x + 4 = 1.6 x - 5 .$

Show solution

Since $3.4 greater than 1.6,$ make the left side the variable side and the right side the constant side.


Subtract $1.6 x$ from both sides.
Combine like terms.
Subtract 4 from both sides.
Simplify.
Use the Division Property of Equality.
Simplify.
Check: Let $x equals −5$ .

Solve Equations Using a General Strategy

So far, we have solved a few specific forms of a linear equation. It’s time now to lay out an overall strategy that can be used to solve any linear equation. We call this the general strategy. Some equations won’t require all the steps to solve, but many will. Simplifying each side of the equation as much as possible first makes the rest of the steps easier.

Example 6

Solve: $3 (x + 2) = 18 .$

Show solution


Simplify each side of the equation as much as possible. Use the Distributive Property.
Collect all variable terms on one side of the equation—all $x$ s are already on the left side.
Collect constant terms on the other side of the equation. Subtract 6 from each side
Simplify.
Make the coefficient of the variable term equal to 1. Divide each side by 3.
Simplify.
Check: Let $x equals 4$ .

Example 7

Solve: $- (x + 5) = 7 .$

Show solution


Simplify each side of the equation as much as possible by distributing. The only $x$ term is on the left side, so all variable terms are on the left side of the equation.
Add 5 to both sides to get all constant terms on the right side of the equation.
Simplify.
Make the coefficient of the variable term equal to 1 by multiplying both sides by -1.
Simplify.
Check: Let $x equals −12$ .

Example 8

Solve: $4 (x - 2) + 5 = −3 .$

Show solution


Simplify each side of the equation as much as possible. Distribute.
Combine like terms
The only $x$ is on the left side, so all variable terms are on one side of the equation.
Add 3 to both sides to get all constant terms on the other side of the equation.
Simplify.
Make the coefficient of the variable term equal to 1 by dividing both sides by 4.
Simplify.
Check: Let $x equals 0$ .

Example 9

Solve: $8 - 2 (3 y + 5) = 0 .$

Show solution

Be careful when distributing the negative.


Simplify—use the Distributive Property.
Combine like terms.
Add 2 to both sides to collect constants on the right.
Simplify.
Divide both sides by −6.
Simplify.
Check: Let $y = - \frac{1}{3}$ .

Example 10

Solve: $3 (x - 2) - 5 = 4 (2 x + 1) + 5 .$

Show solution


Distribute.
Combine like terms.
Subtract $3 x$ to get all the variables on the right since $8 greater than 3$ .
Simplify.
Subtract 9 to get the constants on the left.
Simplify.
Divide by 5.
Simplify.
Check: Substitute: $−4 equals x$ .

In many applications, we will have to solve equations with decimals. The same general strategy will work for these equations.

Example 12

Solve: $0.24 (100 x + 5) = 0.4 (30 x + 15) .$

Show solution


Distribute.
Subtract $12 x$ to get all the $x$ s to the left.
Simplify.
Subtract 1.2 to get the constants to the right.
Simplify.
Divide.	$An image showing the algebraic equation 12x/12 = 4.8/12, which demonstrates how to isolate the variable x by dividing both sides of the equation by 12. The number 12 is colored red below the fraction bar.$
Simplify.
Check: Let $x equals 0.4$ .

Practice Makes Perfect

Solve an Equation with Variables and Constants on Both Sides

In the following exercises, solve the equations for the variable.

$6 x - 15 = 5 x + 3$

$4 x - 17 = 3 x + 2$

Show solution

x = 19

$26 + 8 d = 9 d + 11$

$21 + 6 f = 7 f + 14$

Show solution

f = 7

$3 p - 1 = 5 p - 33$

$8 q - 5 = 5 q - 20$

Show solution

q = −5

$4 a + 5 = - a - 40$

$9 c + 7 = −2 c - 37$

Show solution

c = −4

$8 y - 30 = −2 y + 30$

$12 x - 17 = −3 x + 13$

Show solution

x = 2

$2 z - 4 = 23 - z$

$3 y - 4 = 12 - y$

Show solution

y = 4

$\frac{5}{4} c - 3 = \frac{1}{4} c - 16$

$\frac{4}{3} m - 7 = \frac{1}{3} m - 13$

Show solution

m = −6

$8 - \frac{2}{5} q = \frac{3}{5} q + 6$

$11 - \frac{1}{4} a = \frac{3}{4} a + 4$

Show solution

a = 7

$\frac{4}{3} n + 9 = \frac{1}{3} n - 9$

$\frac{5}{4} a + 15 = \frac{3}{4} a - 5$

Show solution

a = −40

$\frac{1}{4} y + 7 = \frac{3}{4} y - 3$

$\frac{3}{5} p + 2 = \frac{4}{5} p - 1$

Show solution

p = 15

$14 n + 8.25 = 9 n + 19.60$

$13 z + 6.45 = 8 z + 23.75$

Show solution

z = 3.46

$2.4 w - 100 = 0.8 w + 28$

$2.7 w - 80 = 1.2 w + 10$

Show solution

w = 60

$5.6 r + 13.1 = 3.5 r + 57.2$

$6.6 x - 18.9 = 3.4 x + 54.7$

Show solution

x = 23

Solve an Equation Using the General Strategy

In the following exercises, solve the linear equation using the general strategy.

$5 (x + 3) = 75$

$4 (y + 7) = 64$

Show solution

y = 9

$8 = 4 (x - 3)$

$9 = 3 (x - 3)$

Show solution

x = 6

$20 (y - 8) = −60$

$14 (y - 6) = −42$

Show solution

y = 3

$−4 (2 n + 1) = 16$

$−7 (3 n + 4) = 14$

Show solution

n = −2

$3 (10 + 5 r) = 0$

$8 (3 + 3 p) = 0$

Show solution

p = −1

$\frac{2}{3} (9 c - 3) = 22$

$\frac{3}{5} (10 x - 5) = 27$

Show solution

x = 5

$5 (1.2 u - 4.8) = −12$

$4 (2.5 v - 0.6) = 7.6$

Show solution

v = 1

$0.2 (30 n + 50) = 28$

$0.5 (16 m + 34) = −15$

Show solution

m = −4

$- (w - 6) = 24$

$- (t - 8) = 17$

Show solution

t = −9

$9 (3 a + 5) + 9 = 54$

$8 (6 b - 7) + 23 = 63$

Show solution

b = 2

$10 + 3 (z + 4) = 19$

$13 + 2 (m - 4) = 17$

Show solution

m = 6

$7 + 5 (4 - q) = 12$

$−9 + 6 (5 - k) = 12$

Show solution

$k = \frac{3}{2}$

$15 - (3 r + 8) = 28$

$18 - (9 r + 7) = −16$

Show solution

r = 3

$11 - 4 (y - 8) = 43$

$18 - 2 (y - 3) = 32$

Show solution

y = −4

$9 (p - 1) = 6 (2 p - 1)$

$3 (4 n - 1) - 2 = 8 n + 3$

Show solution

n = 2

$9 (2 m - 3) - 8 = 4 m + 7$

$5 (x - 4) - 4 x = 14$

Show solution

x = 34

$8 (x - 4) - 7 x = 14$

$5 + 6 (3 s - 5) = −3 + 2 (8 s - 1)$

Show solution

s = 10

$−12 + 8 (x - 5) = −4 + 3 (5 x - 2)$

$4 (x - 1) - 8 = 6 (3 x - 2) - 7$

Show solution

$x = \frac{1}{2}$

$7 (2 x - 5) = 8 (4 x - 1) - 9$

Corequisite Skills for Introductory Statistics is a custom course edition compiled for Introductory Statistics corequisite courses from OpenStax Contemporary Mathematics, Elementary Algebra 2e, Intermediate Algebra 2e, Prealgebra 2e. Content by OpenStax is reproduced as published, apart from any corrections recorded for this edition. Sections are renumbered to follow the course outline. The course outline is adapted from Larry Green's Support Course for Elementary Statistics (LibreTexts, https://stats.libretexts.org/Bookshelves/Introductory_Statistics/Support_Course_for_Elementary_Statistics), used under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). Lessons 2.4 and 6.7 are written for this edition by XYZ Homework. Every OpenStax lesson is abridged to the subsections and practice exercises this course uses. Lessons are headed with their course titles, not the source section titles. Access the complete originals free at openstax.org. License: CC-BY-NC-SA-4.0.


Distribute.
Add $x$ to get all the variables on the left.
Simplify.
Add 1 to get constants on the right.
Simplify.
Divide by 4.	$A mathematical equation is displayed on a white background. The equation reads '4x/4 = 6/4', where the '4' in the denominator of both fractions is colored red. The '4x' and '6' are in black.$
Simplify.	$The image displays a mathematical equation, x = 3/2, set against a plain white background. The variable 'x' is shown equal to the fraction three halves, with a horizontal line separating the numerator '3' from the denominator '2'.$
Check: Let $x = \frac{3}{2}$ .