Solving Linear Equations

A linear equation is an equation where the variable (usually $x$) appears only to the first power — no exponents, no square roots, no $x$ in a denominator. Solving it means finding the value of the variable that makes the equation true.

The core strategy is simple: use inverse operations to isolate the variable on one side of the equals sign.

Inverse Operations

Every operation has an opposite that undoes it:

The golden rule: whatever you do to one side of the equation, you must do to the other side.

One-Step Equations

These require a single inverse operation.

Example 1: $x + 7 = 15$

Subtract 7 from both sides:

$x + 7 - 7 = 15 - 7$

$x = 8$

Answer: $x = 8$

Example 2: $3x = 21$

Divide both sides by 3:

$\frac{3x}{3} = \frac{21}{3}$

$x = 7$

Answer: $x = 7$

Two-Step Equations

With two operations applied to the variable, undo them in reverse order — handle addition or subtraction first, then multiplication or division.

Example 3: $2x + 5 = 13$

Step 1 — Subtract 5 from both sides:

$2x + 5 - 5 = 13 - 5$

$2x = 8$

Step 2 — Divide both sides by 2:

$\frac{2x}{2} = \frac{8}{2}$

$x = 4$

Answer: $x = 4$

Check: $2(4) + 5 = 8 + 5 = 13$ . Correct.

Multi-Step Equations

When the equation has parentheses or like terms, simplify first, then isolate the variable.

Example 4: $3(x + 4) = 27$

Step 1 — Distribute the 3:

$3x + 12 = 27$

Step 2 — Subtract 12 from both sides:

$3x = 15$

Step 3 — Divide both sides by 3:

$x = 5$

Answer: $x = 5$

Check: $3(5 + 4) = 3(9) = 27$ . Correct.

Variables on Both Sides

When the variable appears on both sides of the equation, move all variable terms to one side and all constant terms to the other.

Example 5: $5x - 3 = 2x + 9$

Step 1 — Subtract $2x$ from both sides to collect variable terms on the left:

$5x - 2x - 3 = 9$

$3x - 3 = 9$

Step 2 — Add 3 to both sides:

$3x = 12$

Step 3 — Divide both sides by 3:

$x = 4$

Answer: $x = 4$

Check: Left side: $5(4) - 3 = 20 - 3 = 17$. Right side: $2(4) + 9 = 8 + 9 = 17$ . Both sides equal 17.

Equations with the Distributive Property

When both sides have expressions to distribute, expand everything first, then combine like terms.

Example 6: $2(3x - 1) = 4(x + 3)$

Step 1 — Distribute on both sides:

$6x - 2 = 4x + 12$

Step 2 — Subtract $4x$ from both sides:

$2x - 2 = 12$

Step 3 — Add 2 to both sides:

$2x = 14$

Step 4 — Divide both sides by 2:

$x = 7$

Answer: $x = 7$

Real-World Application: Electrician — Finding Unknown Current with Ohm’s Law

Ohm’s Law states:

$V = I \times R$

where $V$ is voltage (volts), $I$ is current (amps), and $R$ is resistance (ohms).

An electrician measures 120 volts across a circuit with 15 ohms of resistance. What is the current?

$120 = I \times 15$

Step 1 — Divide both sides by 15:

$\frac{120}{15} = I$

$I = 8$

Answer: The current is 8 amps. The electrician uses this to verify the circuit is within the safe amperage rating for the wire gauge and breaker installed.

Common Mistakes to Avoid

1. Forgetting to apply an operation to both sides. If you subtract 5 from the left, you must subtract 5 from the right.
2. Distributing incorrectly. Remember: $3(x + 4) = 3x + 12$, not $3x + 4$.
3. Sign errors with negatives. Be especially careful when subtracting a negative: $x - (-3) = x + 3$.
4. Not checking your answer. Plugging your solution back into the original equation catches mistakes quickly.

Practice Problems

Test your understanding with these problems. Click to reveal each answer.

Key Takeaways

• A linear equation has the variable to the first power only
• Inverse operations undo what was done to the variable — addition undoes subtraction, multiplication undoes division
• Always perform the same operation on both sides of the equation
• For multi-step equations: distribute first, combine like terms, then isolate the variable
• When variables appear on both sides, move them all to one side first
• Always check your answer by substituting it back into the original equation

Return to Algebra for more topics in this section.