8.2 Solving Linear Equations

The Principle of Inverse Operations

A linear equation in one variable is a statement that two expressions are equal, where the variable appears only to the first power (no x², no square roots). Solving means finding the value of the variable that makes the statement true. The core tool is inverse operations: addition and subtraction undo each other, and multiplication and division undo each other. Whatever you do to one side, you must do to the other to keep the equation balanced.

Think of an equation as a balance scale: if you remove 5 from the left side, you must also remove 5 from the right side.

One-Step Equations

Example 1. Solve x + 6 = 14. Subtract 6 from both sides: x = 8. Check: 8 + 6 = 14. Correct.

Example 2. Solve 4x = 28. Divide both sides by 4: x = 7. Check: 4·7 = 28. Correct.

Example 3. Solve x − 9 = 3. Add 9 to both sides: x = 12. Check: 12 − 9 = 3. Correct.

Two-Step Equations

A two-step equation has the variable both multiplied and added (or subtracted). Undo the addition or subtraction first, then undo the multiplication or division.

Example 4. Solve 2x + 5 = 17.

Step 1: Subtract 5 from both sides. 2x = 12.

Step 2: Divide both sides by 2. x = 6.

Check: 2·6 + 5 = 12 + 5 = 17. Correct.

The order matters. Always peel away the addition or subtraction first, then the multiplication or division. If you divided first in Example 4, you would get x + 2.5 = 8.5, which works but introduces unnecessary decimals.

Multi-Step Equations

A multi-step equation may require combining like terms on one or both sides, distributing to remove parentheses, or moving variables to one side.

Example 5. Solve 3x + 2x − 4 = 16.

Combine like terms on the left first: 5x − 4 = 16. Add 4 to both sides: 5x = 20. Divide by 5: x = 4. Check: 3·4 + 2·4 − 4 = 12 + 8 − 4 = 16. Correct.

Example 6. Solve 5(x − 2) = 2x + 7.

Step 1: Distribute the 5 on the left side: 5x − 10 = 2x + 7.

Step 2: Move the variable terms to one side. Subtract 2x from both sides: 3x − 10 = 7.

Step 3: Move the constant to the other side. Add 10 to both sides: 3x = 17.

Step 4: Divide by 3: x = 17/3, which is about 5.67. You can leave the answer as the fraction 17/3 unless the problem asks for a decimal.

Check: 5(17/3 − 2) = 5(17/3 − 6/3) = 5(11/3) = 55/3. And 2(17/3) + 7 = 34/3 + 21/3 = 55/3. Both sides match.

Equations with Variables on Both Sides

When the variable appears on both sides, collect the variable terms on one side and the constants on the other. It does not matter which side you choose for the variable, but consistency helps you avoid sign mistakes.

Example 7. Solve 7x − 3 = 4x + 9.

Subtract 4x from both sides: 3x − 3 = 9. Add 3 to both sides: 3x = 12. Divide by 3: x = 4. Check: 7·4 − 3 = 28 − 3 = 25, and 4·4 + 9 = 16 + 9 = 25. Correct.

Equations with Parentheses

When parentheses appear, distribute first, then proceed as above.

Example 8. Solve 3(2x − 1) − 4 = 2(x + 5).

Distribute on both sides: 6x − 3 − 4 = 2x + 10. Combine like terms on the left: 6x − 7 = 2x + 10. Subtract 2x: 4x − 7 = 10. Add 7: 4x = 17. Divide by 4: x = 17/4 = 4.25.

Check: Left side = 3(2·4.25 − 1) − 4 = 3(8.5 − 1) − 4 = 3·7.5 − 4 = 22.5 − 4 = 18.5. Right side = 2(4.25 + 5) = 2·9.25 = 18.5. Both sides match.

Equations with Fractions

Fractions in an equation can make the algebra look messy. The clean approach is to multiply every term by the least common denominator (LCD) to clear the fractions first, then solve the resulting integer equation.

Example 9. Solve x/3 + 4 = 9.

The only denominator is 3, so the LCD is 3. Multiply every term by 3: 3·(x/3) + 3·4 = 3·9, giving x + 12 = 27. Subtract 12: x = 15.

Check: 15/3 + 4 = 5 + 4 = 9. Correct.

Example 10. Solve x/2 + x/4 = 6.

The denominators are 2 and 4, so the LCD is 4. Multiply every term by 4: 4·(x/2) + 4·(x/4) = 4·6, giving 2x + x = 24, so 3x = 24 and x = 8.

Check: 8/2 + 8/4 = 4 + 2 = 6. Correct.

Common Traps

• Forgetting to apply an operation to both sides. If you subtract 5 from the left, you must subtract 5 from the right. Skipping one side is the most frequent source of wrong answers.
• Sign errors when moving terms. When you move 2x to the other side by subtraction, you write −2x, not +2x. The sign flips when a term crosses the equals sign.
• Dropping a term. When distributing, multiply every term inside the parentheses. A common error is 3(x + 4) = 3x + 4, where the 4 was never multiplied.
• Forgetting to clear all denominators. When you multiply by the LCD, apply it to every term, including ones without fractions. In x/2 + 3 = 7, multiplying by 2 gives x + 6 = 14, not x + 3 = 14.

Checking Solutions

Substitute your answer back into the original equation—before you simplified—to confirm both sides are equal. This step catches arithmetic mistakes and sign errors. If the two sides do not match, retrace your steps; the error is usually in a sign or a missed term.