Linear equations in one variable

Algebra is about 35% of SAT Math (13–15 of 44 questions). Linear equations in one variable appear in both modules, as multiple choice and as typed answers, often inside a short real-life story.

Keep the balance

An equation is a balance: whatever you do to one side, do to the other. Work in the same order every time. 1) Remove brackets by distributing. 2) Clear fractions or decimals by multiplying every term by the same number. 3) Collect the x-terms on one side and the plain numbers on the other. 4) Divide by the coefficient of x. 5) Check by putting your answer back into the original equation.

One, none or infinitely many solutions

After simplifying, every linear equation looks like ax + b = cx + d. If the x-coefficients are different (a ≠ c), there is exactly one solution. If the x-coefficients are equal but the constants are different (a = c, b ≠ d), the x-terms cancel and you are left with something false, such as 3 = 5: there is no solution. If both sides are exactly the same (a = c and b = d), the equation is an identity: every number is a solution. The SAT likes to ask for the constant that makes an equation have no solution or infinitely many.

See it as two lines

Each side of an equation can be drawn as a line: y = left side and y = right side. The solution is the x-coordinate of the point where the lines cross. Lines with the same slope never cross unless they are the same line, which is why equal x-coefficients mean no solution or infinitely many.

Answer the question that was asked

Many SAT questions do not ask for x. They ask for 2x + 3, or x − 1, or 6x. Before solving, look at the last line of the question. Sometimes you can get the asked expression directly: if 6x + 9 = 24, divide both sides by 3 to get 2x + 3 = 8 without ever finding x.

From words to an equation

Name the unknown with a letter and say what it means, with units. Write each amount in the story as an expression in that letter. Then find the sentence that says two amounts are equal (“costs the same”, “the total was”, “is twice as long as”) and turn it into the equation.

• Losing a minus sign when distributing−2(x − 3) = −2x + 6, not −2x − 6. Multiply the minus by every term inside the brackets.
• Clearing fractions in only some termsWhen you multiply by 12, multiply every term, including the ones without a fraction: x/3 + 1 = 5 becomes 4x + 12 = 60.
• Answering x when the question asks for something elseUnderline what is asked (2x + 3, x − 1, the cost…) and read it again before choosing.
• Mixing up “no solution” and “infinitely many”Same x-coefficient + different constants → none. Everything the same → infinitely many.
• Typing an answer in the wrong formIn typed-answer questions enter 7/2 or 3.5, never 3 1/2. Up to 5 characters for a positive answer, 6 for a negative one. No $ or % signs and no units.