Linear equations in one variable
These are the most common questions on the SAT Math section. Most are a few steps of careful algebra — the points are usually lost to small mistakes, misreading what's asked, or missing a "no solution" setup.
What College Board tests
Solving an equation with one variable, often after expanding parentheses or clearing fractions. Many questions set the equation inside a real-world situation, ask for the value of an expression rather than the variable, or test whether an equation has one solution, no solution, or infinitely many.
The method
Every linear equation is solved by the same sequence. Do the steps in order and most of these become routine.
One, none, or infinitely many
When the variable terms are identical on both sides, the variable cancels. What's left tells you the answer: a true statement like means infinitely many solutions; a false statement like means no solution.
Four worked examples in SAT format. Work each one before tapping Show the answer.
1 · Solve within a situation
A technician charges a flat fee of $35 for a house call, plus $8 for each quarter-hour of work. The total charge for one visit was $131. The equation represents this situation, where is the number of quarter-hours worked. How many quarter-hours did the technician work?
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Answer: B
Subtract the flat fee from both sides, then divide by the per-unit rate:
A Divides 131 by 8 and ignores the flat fee.
C Adds the 35 instead of subtracting: rounded.
D Results from a setup error in isolating .
2 · Find the value of an expression
If , what is the value of ?
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Answer: B
You don't need at all. The quantity is multiplied by 3, so divide both sides by 3:
Recognizing that the question asks for the whole expression saves several steps.
A Solves all the way to — the value of , not .
C Adds 3 to 11 instead of recognizing the expression is already found.
D Results from distributing incorrectly before dividing.
3 · One, none, or infinitely many
In the equation , is a constant. For what value of does the equation have no solution?
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Answer: C
An equation has no solution when the variable terms are identical on both sides but the constants differ — the variable cancels and leaves a false statement. Setting gives . Subtracting from both sides leaves , which is never true. So there is no solution.
A Gives , which has exactly one solution.
B Gives , which has one solution.
D Matches a constant, not the coefficient; the equation still has one solution.
4 · Translate a percentage into an equation
After a 20% discount, the price of a jacket is $52. What was the original price, in dollars, before the discount?
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Answer: C
A 20% discount means the price paid is 80% of the original. If is the original price, then . Divide both sides by 0.80:
A Takes 20% of 52 and subtracts — discounting the sale price again.
B Adds 20% of 52 to 52, treating the discount as applied to the wrong base.
D Divides by a different factor; is 65, not 72.