Linear Equations, Inequalities & Systems
The largest single category on SAT Math. Master isolating a variable, solving systems by substitution/elimination, and translating word problems into equations.
Solving and translating
Isolate the variable using inverse operations, applying the same operation to both sides. For systems of two equations, substitution (solve one equation for a variable, plug into the other) and elimination (add/subtract equations to cancel a variable) are both fast on the SAT — pick whichever avoids fractions.
For word problems, define a variable explicitly, then translate each phrase into an equation piece by piece ('is' means =, 'more than' means +, 'times' means ×) before solving.
For a system with no solution, the two lines are parallel (same slope, different intercepts). For infinitely many solutions, the two equations are the same line (proportional coefficients including the constant).
Worked Example
Solve the system: 2x + y = 10 and x - y = 2.
- Add the two equations to eliminate y: (2x+y) + (x-y) = 10+2 → 3x = 12 → x = 4.
- Substitute x=4 into x-y=2: 4-y=2 → y=2.
Answer: x=4, y=2
Key terms (1)
- Elimination method
- Adding or subtracting two equations to cancel one variable.
Practice Quiz
If 3x - 7 = 2x + 5, what is the value of x?