Linear equations in one variable
Balance scales, undo moves and a few clever shortcuts. By the end you will solve any one-variable equation the SAT throws at you, and spot the traps built into the wrong answers.
An equation is a balance
The equals sign means both sides weigh the same. Each gold box is one x and each teal ball is 1. Your job is to get a single x alone on the left while keeping the scale level. The rule: whatever you do to one side, you do to the other.
Undo in reverse order
To build 4x − 7, you start with x, multiply by 4, then subtract 7. To solve, undo those moves in reverse: add 7 first, then divide by 4. Think of taking off shoes before socks.
- The last thing done to x was “subtract 7”. Undo it: add 7 to both sides. 4x = 28
- Next, undo “multiply by 4”: divide both sides by 4. x = 7
- Check by substituting: 4(7) − 7 = 28 − 7 = 21. It works.
Variables on both sides
When x appears on both sides, collect the x terms on one side and the numbers on the other. A handy habit: subtract the smaller x term, so you keep a positive coefficient.
- Subtract 3x from both sides: 4x − 4 = 12
- Add 4 to both sides: 4x = 16
- Divide by 4: x = 4. Check: 7(4) − 4 = 24 and 3(4) + 12 = 24. Both sides match.
Brackets and fractions
Two quick clean-ups make almost any equation look like the ones you already know. Brackets: multiply the outside number by every term inside. Fractions: multiply every term on both sides by the denominator (or the lowest common denominator) and the fractions disappear.
- Expand: 3x − 12 = 2x + 1
- Subtract 2x: x − 12 = 1
- Add 12: x = 13
- The lowest common denominator is 6. Multiply every term by 6: 3x + 2x = 60
- Combine: 5x = 60
- Divide by 5: x = 12
Common trapMultiplying only the fraction and forgetting the other side. In x/2 + 4 = 9, multiplying by 2 gives x + 8 = 18, not x + 4 = 18.
One, none or infinitely many
Most equations have exactly one answer. Two special cases come up on the SAT every year. Simplify both sides and look at the x terms:
- Different x coefficients (like 4x and 2x): exactly one solution.
- Same x coefficients, different numbers (like 2x + 3 = 2x + 5): the x terms cancel and you get 3 = 5, which is never true. No solution.
- Both sides identical (like 3x + 6 = 3x + 6): every x works. Infinitely many solutions.
Tap a card, then tap the box it belongs in.
Seeing structure: the shortcut
The SAT loves asking for an expression rather than x. If the expression is a multiple of something you already know, skip solving entirely. It saves time and removes chances to slip.
From words to equations
A reliable three-step method: 1. Name the unknown with a letter. 2. Write the relationship in words (“fee plus rate times hours equals total”). 3. Swap in the numbers and solve. Then reread the question to be sure you answer what it asks.
Spot the trap
SAT wrong answers are built from real slips. Each solution below has one faulty step. Find it and you will recognise it on test day.
Check your answer with Desmos
The Digital SAT includes the Desmos calculator. For clean one-variable equations, solving by hand is quicker, but Desmos is a great way to confirm an answer or to see why an equation has no solution. Press play on each move.