Solving linear equations

How to solve equations such as 3x + 5 = 20 by undoing operations, and how to check your answer.

An equation states that two expressions are equal, for example $3 x + 5 = 20$ . Solving it means finding the value of the variable that makes the statement true. An equation is linear when the variable appears only to the first power (no $x^{2}$ , no $x$ , no $x$ in a denominator).

The balance idea

Think of an equation as a balanced scale. If you do the same thing to both sides, the scale stays balanced. So you may:

add or subtract the same number on both sides,
multiply or divide both sides by the same non-zero number.

The goal is to get the variable alone on one side, by undoing what was done to it in reverse order.

One-step and two-step equations

Example 1. Solve $x + 7 = 12$ .

Subtract 7 from both sides: $x = 5$ .

Example 2. Solve $3 x + 5 = 20$ .

$x$ was multiplied by 3, then 5 was added. Undo in reverse: subtract 5, then divide by 3.

$3 x = 15 ⟹ x = 5.$

Check. Put $x = 5$ back into the original equation: $3 \cdot 5 + 5 = 20$ . ✓ Always check; it takes seconds and catches sign errors.

Variables on both sides

Collect the variable terms on one side and the numbers on the other.

Example 3. Solve $5 x - 4 = 2 x + 11$ .

$5 x - 2 x 3 x x = 11 + 4 = 15 = 5.$

Brackets and fractions

First remove brackets (distributive property), then collect terms. For fractions, multiply both sides by the common denominator to clear them.

Example 4. Solve $2 (x - 3) = 10$ .

$2 x - 6 = 10 \Rightarrow 2 x = 16 \Rightarrow x = 8.$

Example 5. Solve $\frac{x}{2} + \frac{x}{3} = 10$ .

The common denominator is 6, so multiply both sides by 6:

$3 x + 2 x = 60 \Rightarrow 5 x = 60 \Rightarrow x = 12.$

How many solutions?

A linear equation in one variable has usually exactly one solution, but two other outcomes exist:

$2 x + 1 = 2 x + 5$ simplifies to $1 = 5$ , which is never true: no solution.
$2 (x + 1) = 2 x + 2$ simplifies to $2 = 2$ , which is always true: infinitely many solutions (every $x$ works).

Word problems

Name the unknown with a variable.
Translate the sentences into an equation.
Solve it, then check against the words, not just the equation.

Example 6. A phone plan costs 10 per month plus 0.05 per minute. Your bill was 16.50. How many minutes did you talk?

Let $m$ be the minutes: $10 + 0.05 m = 16.5$ , so $0.05 m = 6.5$ and $m = 130$ .

Common mistakes

Changing only one side of the equation.
Sign errors when moving a term: moving $- 4$ to the other side makes it $+ 4$ (you added 4 to both sides).
Dividing only part of one side: in $3 x + 6 = 12$ , dividing by 3 gives $x + 2 = 4$ , not $x + 6 = 4$ .
Forgetting to multiply every term when clearing fractions.

Practice

Solve $4 x - 9 = 15$ .
Solve $7 - 2 x = 3 x + 22$ .
Solve $3 (2 x + 1) = 4 x + 13$ .
Solve $\frac{x + 1}{3} = \frac{x - 1}{2}$ .
Three consecutive integers add up to 72. Find them.

Answers

$x = 6$ .
$- 15 = 5 x$ , so $x = - 3$ .
$6 x + 3 = 4 x + 13 \Rightarrow 2 x = 10 \Rightarrow x = 5$ .
Multiply by 6: $2 (x + 1) = 3 (x - 1) \Rightarrow 2 x + 2 = 3 x - 3 \Rightarrow x = 5$ .
$n + (n + 1) + (n + 2) = 72 \Rightarrow 3 n = 69 \Rightarrow n = 23$ : the numbers are 23, 24, 25.

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