Linear Equations
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.
A linear equation is one where the variable appears to the first power only — no , no , no . Examples: , , .
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: .
- 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 terms to one side.
- Move all constants to the other side.
- Divide by the coefficient on .
Special cases to recognize:
- No solution (sometimes phrased "the equation has no solutions"): when isolating leads to something like — a false statement. The original equation is impossible.
- Infinitely many solutions: when isolating leads to or — always true. Every value of works.
A common SAT twist: they give you an equation like and ask for what value of the equation has no solution. To get no solution in a linear equation, you need the coefficients to match (so the 's cancel) AND the constants to differ. So: → no solution if you can rearrange to make (nonzero). In fact, this works only for the specific structure where coefficients aren't equal — for infinitely many, you need the coefficients AND constants to match.
| 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 " is a solution to the equation," it usually means is one specific value. Solve as normal.
Quick check. Distribute, combine, move x's left, constants right, divide. Plug your answer back in to verify.
If 6x + 15 = 33, what is the value of x?
Worked examples
Solve for :
For what value of does the equation have no solution?
Common pitfalls
becomes , NOT . Distribute the negative sign to every term inside the parentheses. This is the most common arithmetic slip on SAT linear questions.
Cross-multiplication only works between two fractions. is not solved by 'cross-multiplying' — just multiply both sides by 3 to get .
IS a solution (the value 0). 'No solution' means there's no number that makes the equation true. If you arrive at when solving, that's no solution — not zero.
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.
Key takeaways
Solving a linear equation = isolating the variable. Do the same operation to both sides.
Standard order: distribute, combine, move 's left, move constants right, divide.
'No solution' happens when the equation reduces to a false statement (like ).
'Infinitely many solutions' happens when it reduces to a true statement (like ).
Always plug your answer back in — it catches the most common arithmetic slips.
Watch & learn
Curated Khan Academy walkthroughs on Linear Equations. They're complementary to this lesson — watch one if a written explanation isn't clicking, or after to reinforce.
Try it yourself
5 practice questions on Linear Equations, drawn from the question bank. The tutor is one click away if you get stuck.