Math

Linear Equations in 1 Variable

2 min readEasy5-question drill

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.

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→Linear Equations in 2 Variables →Linear Inequalities in 1 or 2 Variables →Systems of 2 Linear Equations in 2 Variables

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.

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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?

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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.

Watch & learn

Curated Khan Academy walkthroughs on Linear Equations in 1 Variable. They're complementary to this lesson — watch one if a written explanation isn't clicking, or after to reinforce.

Tracks your progress across lessons.

Try it yourself

5 practice questions on Linear Equations in 1 Variable, drawn from the question bank. The tutor is one click away if you get stuck.

Where this lesson fits

Linear Equations in 1 Variable

You are here

→Linear Equations in 2 Variables →Linear Inequalities in 1 or 2 Variables →Systems of 2 Linear Equations in 2 Variables