Linear equations are among the most fundamental topics in algebra. Once you understand the process for solving them, you can tackle everything from homework problems to real-world applications. This guide walks you through one-step, two-step, and multi-step equations with clear worked examples at every stage.
A linear equation is any equation where the variable has an exponent of 1. That means the variable is never squared, cubed, or raised to any higher power. The standard form looks like ax + b = c, where a, b, and c are numbers and x is the variable you want to find.
The goal when solving a linear equation is always the same: isolate the variable on one side of the equals sign so you end up with a statement like x = (some number). You do this by performing inverse operations — undoing whatever has been done to the variable.
One-step equations can be solved in a single operation. Identify what is being done to the variable, then apply the inverse operation to both sides of the equation.
Example: x + 7 = 12
Subtract 7 from both sides: x + 7 − 7 = 12 − 7 → x = 5
Example: x − 3 = 10
Add 3 to both sides: x − 3 + 3 = 10 + 3 → x = 13
Example: 4x = 28
Divide both sides by 4: 4x ÷ 4 = 28 ÷ 4 → x = 7
The key idea is balance: whatever you do to one side, you must do to the other. This keeps the equation true.
Two-step equations require two operations to isolate the variable. The general strategy is to undo addition or subtraction first, then undo multiplication or division. Think of it as peeling away layers in reverse order.
Example: 2x + 5 = 13
Step 1 — Subtract 5 from both sides: 2x + 5 − 5 = 13 − 5 → 2x = 8
Step 2 — Divide both sides by 2: 2x ÷ 2 = 8 ÷ 2 → x = 4
Example: 3x − 7 = 20
Step 1 — Add 7 to both sides: 3x − 7 + 7 = 20 + 7 → 3x = 27
Step 2 — Divide both sides by 3: 3x ÷ 3 = 27 ÷ 3 → x = 9
Always handle the addition or subtraction step before tackling multiplication or division. This simplifies the equation more quickly and avoids fractions.
When the variable appears on both sides of the equation, start by moving all variable terms to one side and all constant terms to the other. Then solve as you would a two-step equation.
Example: 5x + 3 = 2x + 15
Subtract 2x from both sides: 5x − 2x + 3 = 15 → 3x + 3 = 15
Subtract 3 from both sides: 3x = 12
Divide both sides by 3: x = 4
Example: 4x − 1 = 2x + 9
Subtract 2x from both sides: 4x − 2x − 1 = 9 → 2x − 1 = 9
Add 1 to both sides: 2x = 10
Divide both sides by 2: x = 5
Tip: it does not matter which side you move the variables to. Pick whichever side keeps the coefficient positive to avoid working with negative numbers.
After solving an equation, always substitute your answer back into the original equation to verify it works. If both sides are equal, your solution is correct.
Example: Check x = 4 for 5x + 3 = 2x + 15
Left side: 5(4) + 3 = 20 + 3 = 23
Right side: 2(4) + 15 = 8 + 15 = 23
Both sides equal 23, so x = 4 is correct.
This step only takes a few seconds and can save you from avoidable mistakes on tests and assignments. Make it a habit.
Many real-world problems can be translated into linear equations. The key is to define your variable, write the equation, and then solve.
Example: A number doubled and increased by 5 equals 21. Find the number.
Let x = the number. The equation is 2x + 5 = 21.
Subtract 5: 2x = 16. Divide by 2: x = 8.
Example: Maria has 3 times as many stickers as Tom. Together they have 48. How many does Tom have?
Let x = the number of stickers Tom has. Maria has 3x. Together: x + 3x = 48.
Combine like terms: 4x = 48. Divide by 4: x = 12. Tom has 12 stickers.
When tackling word problems, read carefully, identify the unknown, and translate the words into math one phrase at a time.
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