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Linear Equations

4 min readEasy10-question checkpoint

Linear equations — the *solve for x* questions — are the bedrock of the SAT Math section. Master them and you unlock every later topic that builds on them.

Where this lesson fits

Linear Equations

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→Linear Inequalities →Linear Functions →Systems of Equations

A linear equation is one where the variable appears to the first power only — no $x^2$ , no $\sqrt{x}$ , no $\frac{1}{x}$ . Examples: $3x + 7 = 22$ , $5(x-2) = 4x + 1$ , $\frac{x}{4} - 3 = 7$ .

The goal is always the same: isolate the variable. Whatever you do to one side of the equation, do to the other. The four moves you'll use:

Add or subtract the same quantity from both sides.
Multiply or divide both sides by the same nonzero number.
Distribute when there are parentheses: $5(x-2) = 5x - 10$ .
Combine like terms on each side before isolating.

The standard order of operations to solve:

Distribute any parentheses.
Combine like terms on each side.
Move all $x$ terms to one side.
Move all constants to the other side.
Divide by the coefficient on $x$ .

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Solving any linear equation — same playbook

Are there parentheses anywhere?

Yes ↓

Distribute. Then: are there like terms on either side?

Yes ↓

→ Combine, then move x's to one side, constants to the other, then divide

No ↓

→ Move x's to one side, constants to the other, then divide

No ↓

Are there like terms on either side?

Yes ↓

→ Combine, then move x's to one side, constants to the other, then divide

No ↓

→ Move x's to one side, constants to the other, then divide

Special cases to recognize:

No solution (sometimes phrased "the equation has no solutions"): when isolating leads to something like $5 = 7$ — a false statement. The original equation is impossible.
Infinitely many solutions: when isolating leads to $0 = 0$ or $x = x$ — always true. Every value of $x$ works.

A common SAT twist: they give you an equation like $3x + b = 5x - 7$ and ask for what value of $b$ the equation has no solution. To get no solution in a linear equation, you need the $x$ coefficients to match (so the $x$ 's cancel) AND the constants to differ. So: $3x + b = 5x - 7$ → no solution if you can rearrange to make $0 =$ (nonzero). In fact, this works only for the specific structure where $x$ coefficients aren't equal — for infinitely many, you need the coefficients AND constants to match.

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Linear equation outcomes — three possibilities

After simplifying, you get…	Type	Meaning
$x = $ a number (e.g. $x = 5$)	One solution	The unique value that makes the equation true
A false statement (e.g. $5 = 7$)	No solution	Impossible — no number satisfies it
A true statement (e.g. $0 = 0$, $x = x$)	Infinite solutions	Every value of $x$ works

SAT-specific: when you see " $x$ is a solution to the equation," it usually means $x$ is one specific value. Solve as normal.

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Quick check

Quick check. Distribute, combine, move x's left, constants right, divide. Plug your answer back in to verify.

An online store charges $8.50 for shipping plus$ 1.25 per pound for package weight. If the total shipping cost was $21.00, how many pounds did the package weigh?

Worked examples

Example 1

Solve for $x$ :

3(x - 4) + 5 = 2x + 1

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Example 2

For what value of $b$ does the equation $4x + b = 4x + 9$ have no solution?

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Common pitfalls

Forgetting to distribute negatives

$-(2x - 5)$ becomes $-2x + 5$ , NOT $-2x - 5$ . Distribute the negative sign to every term inside the parentheses. This is the most common arithmetic slip on SAT linear questions.

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Cross-multiplying a single fraction with a single number

Cross-multiplication only works between two fractions. $\frac{x}{3} = 5$ is not solved by 'cross-multiplying' — just multiply both sides by 3 to get $x = 15$ .

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Confusing 'no solution' with $x = 0$

$x = 0$ IS a solution (the value 0). 'No solution' means there's no number that makes the equation true. If you arrive at $5 = 7$ when solving, that's no solution — not zero.

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Skipping the check

The SAT puts trap answers near the right one. Plugging in your answer takes 5 seconds and catches sign errors that would otherwise cost the question. Always check on linear-equation problems.

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Key takeaways

Solving a linear equation = isolating the variable. Do the same operation to both sides.
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Standard order: distribute, combine, move $x$ 's left, move constants right, divide.
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'No solution' happens when the equation reduces to a false statement (like $5 = 7$ ).
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'Infinitely many solutions' happens when it reduces to a true statement (like $0 = 0$ ).
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Always plug your answer back in — it catches the most common arithmetic slips.
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Checkpoint

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.

Start checkpoint · 10 questions

Where this lesson fits

Linear Equations

You are here

→Linear Inequalities →Linear Functions →Systems of Equations

Worked examples

Common pitfalls

Key takeaways

Further reading

Checkpoint