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Linear Equations in 1 Variable

5 min readEasy10-question skill check

Linear equations in one variable are the most common math you'll see on this test — and the foundation for almost everything else. Master the 'undo' process and you'll bank points fast.

Where this lesson fits

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→Linear Equations in 2 Variables →Equivalent Expressions

A linear equation in one variable is an equation where you're solving for a single unknown (usually x), and that variable is never raised to a power, never multiplied by another variable, and never stuck inside a square root. It's just x to the first power. Examples: 3x + 11 = 32 or 5(x - 2) = 20.

Your goal is always the same: get the variable by itself on one side of the equals sign. The number you end up with is the solution — the value that makes the equation true.

The golden rule: whatever you do to one side, you must do to the other. The equals sign means both sides are balanced, like a scale. If you add 4 to the left, add 4 to the right. If you divide the left by 3, divide the right by 3.

To isolate x, you undo operations in reverse order. Think about what's been done to x:

First undo addition or subtraction (the terms being added or removed).
Then undo multiplication or division (the coefficient stuck to x).

Here's the pattern with 3x + 11 = 32:

Subtract 11 from both sides: 3x = 21
Divide both sides by 3: x = 7

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Is a number added or subtracted next to x?

Yes ↓

→ Undo it first (add/subtract from both sides)

No ↓

→ Divide both sides by the coefficient of x

Order of operations for isolating x.

That's it. Two moves.

If the equation has parentheses, distribute first: 5(x - 2) = 20 becomes 5x - 10 = 20. If the variable appears on both sides, move all the x terms to one side and all the plain numbers to the other:

6x + 3 = 2x + 19 → subtract 2x: 4x + 3 = 19 → subtract 3: 4x = 16 → divide by 4: x = 4.

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Solving 6x + 3 = 2x + 19

Step	Action	Result
1	Subtract 2x from both sides	4x + 3 = 19
2	Subtract 3 from both sides	4x = 16
3	Divide both sides by 4	x = 4

Collecting variable terms when x appears on both sides.

Word problems are linear equations in disguise. "Pens cost $4 each; how many can you buy with $48?" is just 4x = 48, so x = 12. The skill is translating the sentence into an equation, then solving normally.

Always check by plugging your answer back in. If x = 7 in 3x + 11, you get 21 + 11 = 32. ✓ That match confirms you're right.

Quick check

Check your understanding with a question from this topic:

Pens cost $4 each. How many pens can be purchased with$ 48?

Your answer:

Enter a whole number, fraction (e.g. 3/4), or decimal (e.g. .75).

Worked examples

Example 1

Solve for x: 5x - 9 = 26

Example 2

If 4(x + 3) = 2x + 22, what is the value of x?

Example 3

A taxi charges a flat fee of $3 plus $2 for every mile driven. If a ride costs $19, how many miles was the ride?

Common pitfalls

Only changing one side of the equation

If you subtract a number from the left, you MUST subtract it from the right too. Forgetting to balance both sides is the single most common error — treat the equals sign like a scale that must stay level.

Forgetting to distribute to every term

In 5(x - 2), students multiply the 5 by x but forget the -2, writing 5x - 2 instead of 5x - 10. Distribute to everything inside the parentheses.

Dividing too early

In 3x + 11 = 32, don't divide by 3 before subtracting 11 — you'd have to divide every term. Clear the added/subtracted number first, then divide by the coefficient.

Sign errors when moving terms

Moving a term across the equals sign flips its sign. Better yet, don't 'move' — actually add or subtract from both sides so you never lose track of a negative.

Key takeaways

Goal: isolate the variable using inverse operations on both sides.
Undo addition/subtraction first, then multiplication/division.
Distribute through parentheses before collecting terms.
Put all variable terms on one side, all constants on the other.
Always plug your answer back in to confirm it works.

Tracks your progress across lessons.

Skill check

10 questions on Linear Equations in 1 Variable, from our question bank. Score 70% or better across 10 answers to reach Silver and pass this step.

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Where this lesson fits

Linear Equations in 1 Variable

You are here

→Linear Equations in 2 Variables →Equivalent Expressions