Why linear math runs the SAT
Roughly a third of the digital SAT Math section is linear in some form: equations in one or two variables, inequalities, systems, and word problems that hide a linear setup behind a story. Master this and you've stabilized a huge chunk of your score.
The shape that does most of the work: y = mx + b
Almost every linear question lives somewhere in the family of y = mx + b:
- m is the rate of change (slope, price per item, miles per hour).
- b is the starting value (initial cost, beginning population).
- x is what's varying (months, items, hours).
- y is the total or output.
Train yourself to read every linear word problem and ask: What's the rate? What's the start? What varies? Once you've labeled those three, the equation almost writes itself.
Slope: rise over run, but really
Slope is just change in y divided by change in x. Two ways it shows up on the SAT:
- Two points given. Slope = (y₂ - y₁) / (x₂ - x₁). That's it.
- Word problem. Look for "per". Per hour, per dollar, per box. The number attached to "per" is your slope.
Parallel lines have the same slope. Perpendicular lines have slopes that are negative reciprocals (multiply to -1). The SAT loves to test these in disguise: "Line p passes through (2, 5) and is perpendicular to a line with slope 4. What's its equation?" The perpendicular slope is -1/4.
The forms of a line
Three you'll see, all the same line in different costumes:
- Slope-intercept: y = mx + b. Slope and y-intercept ready to read.
- Standard: Ax + By = C. Easy to find x and y intercepts: set the other variable to 0.
- Point-slope: y - y₁ = m(x - x₁). Useful when given a point and a slope.
If a question gives you one form and asks something natural for another, just switch forms. There's no wrong answer for "which form is best"; pick the one that puts the answer in your face.
Solving systems: three reliable moves
A system is two equations, two variables. The SAT loves these. Three approaches:
- Substitution. Solve one equation for one variable, plug into the other. Best when one equation is already nearly solved.
- Elimination (combination). Add or subtract the equations to cancel a variable. Best when the equations line up tidily.
- Graphing on the Bluebook calculator. Type both equations into Desmos and read the intersection. Almost always faster than algebra. Use it.
One special case the SAT keeps testing: "how many solutions?" A system has:
- One solution if the lines have different slopes.
- No solutions if the lines are parallel (same slope, different y-intercepts).
- Infinite solutions if the lines are the same line (one is a multiple of the other).
Test idea: rewrite both equations in y = mx + b form and compare slopes and intercepts. Done.
Inequalities: the same rules, with one twist
Inequalities behave like equations except for one rule: multiplying or dividing both sides by a negative number flips the sign. -2x > 6 becomes x < -3, not x > -3.
System-of-inequalities questions are visual. The Bluebook Desmos handles these brilliantly. Type both inequalities, see the overlapping shaded region, count lattice points if asked, or read the corner of the feasible region for optimization.
Word problems: the translation table
The English-to-math dictionary you actually need:
| English | Math |
|---|---|
| "is", "equals", "totals" | = |
| "more than", "increased by", "plus" | + |
| "less than", "decreased by", "minus" | - (watch order!) |
| "of" (as in "20% of x") | × |
| "per", "for each" | rate (slope) |
| "at most", "no more than" | ≤ |
| "at least", "no less than" | ≥ |
Watch "less than": "5 less than x" is x - 5, not 5 - x.
A worked example
"A taxi charges a $3.50 base fare plus $2.10 per mile. A second taxi charges a $5.00 base fare plus $1.85 per mile. After how many miles do the two taxis cost the same?"
Set up: 3.50 + 2.10m = 5.00 + 1.85m.
Subtract 1.85m: 3.50 + 0.25m = 5.00.
Subtract 3.50: 0.25m = 1.50.
Divide: m = 6.
Or, faster, type both lines into Desmos and read the intersection: x = 6. Either way, six miles.
What to practice this week
- Get fluent in Desmos. The Bluebook app has Desmos built in, and on linear systems it's faster than paper algebra. Practice typing equations until it's reflex.
- Drill word problems by labeling rate, start, variable, total before writing any equation. The setup mistakes vanish.
- For "how many solutions" questions, get fast at converting both equations to y = mx + b form. It's the whole game.