The backbone of SAT Algebra
Linear relationships appear more than any other math topic. A linear equation has variables only to the first power and graphs as a straight line: y = mx + b, where m is the slope and b is the y-intercept.
Solving one equation
Isolate the variable using inverse operations, doing the same thing to both sides. Example: 4x - 7 = 9 → 4x = 16 → x = 4.
Slope and meaning
Slope = change in y over change in x = (y2 - y1) / (x2 - x1). In a word problem, slope is a rate (dollars per hour, miles per gallon) and the intercept is a starting value.
Systems of two equations
You can solve by:
- •Substitution — solve one equation for a variable, plug into the other.
- •Elimination — add or subtract equations to cancel a variable.
- •Graphing in Desmos — the intersection point is the solution.
A system has no solution when the lines are parallel (same slope, different intercept) and infinitely many when the equations are multiples of each other.
Worked example
Solve the system: 2x + y = 11 x - y = 1 Add the equations to eliminate y: 3x = 12, so x = 4. Then 4 - y = 1, so y = 3. Solution: (4, 3). Check: 2(4) + 3 = 11. Correct.
Word-problem tip
Define variables in writing first. "A gym charges a 15 per month" becomes C = 15m + 20. The slope 15 is the monthly rate; 20 is the one-time fee.
Key takeaways
- •Same operation, both sides — always.
- •Elimination is fastest when coefficients line up; substitution when one variable is already isolated.
- •Parallel lines = no solution; identical lines = infinite solutions.
- •Translate rates to slope and fixed amounts to the intercept.
