Algebra is one of the two largest domains on SAT Math — 13–15 of the 44 questions (about 35%), tied with Advanced Math: linear equations, systems, inequalities, and interpreting linear relationships. At the 1500+ level, the content is rarely hard — speed and accuracy are what separate scores.
Golden rule of systems: if one variable is already isolated (or easy to isolate), substitute. If the equations line up neatly, add/subtract to eliminate. Pick the faster path per problem.
What to have automatic
- Slope = (y₂ − y₁)/(x₂ − x₁); slope-intercept form y = mx + b, where b is the y-value at x = 0.
- Solutions of a linear system = the point where the lines meet. No solution = parallel (same slope, different intercept). Infinitely many = the same line.
- When you multiply or divide an inequality by a negative, flip the sign. This is the #1 careless error.
Word-problem move: Define the variable in words first ("let m = number of months"), then translate each sentence into one equation. Set expressions equal when the problem says "the same."
Worked examples
Solve: 5x + 3 = 2x + 18
- Get the variables on one side: subtract 2x from both sides → 3x + 3 = 18.
- Get the numbers on the other side: subtract 3 from both sides → 3x = 15.
- Divide both sides by 3 → x = 5.
- Check: 5(5) + 3 = 28 and 2(5) + 18 = 28. Both sides match, so x = 5 is correct.
A line passes through (2, 3) and (6, 11). Write its equation in slope-intercept form.
- Find the slope: m = (11 − 3) / (6 − 2) = 8/4 = 2.
- Use y = mx + b with one point. Using (2, 3): 3 = 2(2) + b.
- Solve for b: 3 = 4 + b, so b = −1.
- The equation is y = 2x − 1. Check with the other point: 2(6) − 1 = 11. ✓
Solve the inequality: 7 − 3x ≥ 22
- Subtract 7 from both sides: −3x ≥ 15.
- Divide both sides by −3. Dividing by a negative FLIPS the inequality sign.
- x ≤ −5.
- Check with x = −6: 7 − 3(−6) = 25, and 25 ≥ 22 ✓.
Common mistakes to avoid
- Forgetting to flip the inequality sign when you multiply or divide by a negative number. This is the single most common algebra error on the SAT.
- Mixing up rise and run in the slope formula. Slope is change in y over change in x, not the other way around.
- Distributing a negative incorrectly: −2(x − 5) is −2x + 10, not −2x − 10.
- Answering the wrong question. If the problem asks for 3x and you solved x = 4, the answer is 12, not 4. Always re-read what was asked.