Systems of Equations and Inequalities
Systems of Equations and Inequalities
Items here ask you to solve two-variable systems by substitution, elimination, or graphing, classify a system by its number of solutions, and graph the solution set of a system of linear inequalities. Context problems involving mixtures, ticket revenue, or break-even analysis are common.
Core rules
- Substitution: solve one equation for a variable, substitute into the other, solve, then back-substitute to find the other value.
- Elimination: multiply one or both equations so a variable's coefficients become opposites, then add. Multiply every term, including the constant term.
- Classify by slopes: different slopes give exactly one solution (consistent, independent); identical slopes with different intercepts give no solution (inconsistent); identical lines give infinitely many solutions (dependent).
- For inequalities, graph the boundary line dashed for < or > and solid for ≤ or ≥, then test a point to choose the shading. A system's solution is the overlap of the shaded regions.
- A solution must satisfy every equation or inequality in the system at the same time.
Worked example
Solve 2x + 3y = 12 and x − y = 1.
From the second equation, x = y + 1. Substituting: 2(y + 1) + 3y = 12 → 2y + 2 + 3y = 12 → 5y = 10 → y = 2. Then x = 2 + 1 = 3. Check in the first equation: 2(3) + 3(2) = 6 + 6 = 12. ✓ The solution is (3, 2).
Traps that cost marks
- Multiplying only one side of an equation during elimination.
- Sign errors when subtracting one equation from another - adding the opposite is safer.
- Shading the wrong side of a boundary; test (0, 0) whenever it is not on the line.
- Assuming a single solution without comparing slopes.
- Giving a single point as the answer to a system of inequalities, whose solution is a region.