Systems & inequalities

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Two ways to solve

Substitution (best when one equation already isolates a variable): if y = 2x + 1 and 3x + y = 11, substitute: 3x + (2x + 1) = 11 → 5x = 10 → x = 2, y = 5.

Elimination (best when coefficients line up): add or subtract the equations to kill a variable. 2x + 3y = 12 and 2x − y = 4 → subtract: 4y = 8 → y = 2, x = 3. Multiply an equation first if needed to make coefficients match.

On the real test there's a third way — graph both in Desmos and tap the intersection. Learn the algebra anyway: some questions ask about the setup, not the point.

No solution / infinite solutions

Write both lines as y = mx + b and compare:

• Different slopes → exactly one solution (they must cross).
• Same slope, different intercepts → parallel, no solution.
• Same slope, same intercept → same line, infinitely many.

Example: for what value of k does 6x + ky = 7 and 3x + 2y = 5 have no solution? Match slopes: −6/k = −3/2 → k = 4. (Check intercepts differ: 7/4 ≠ 5/2 ✓.)

Inequalities: one extra rule, one graph skill

Solve them exactly like equations, with one exception: multiplying or dividing by a negative flips the inequality sign. −2x > 8 → x < −4. On graphs, ≤/≥ draw solid boundary lines, </> draw dashed ones, and the solution region for a system of inequalities is where the shadings overlap. Test the point (0, 0) to check which side is shaded — if it satisfies the inequality, shade its side.

Common traps

• Answering x when the question asks for x + y — systems questions do this constantly.
• Forgetting the sign flip when dividing an inequality by a negative.
• Word setups: "adult tickets a, child tickets c: a + c = 200 and 12a + 7c = 1900" — one equation counts things, the other counts money. Don't mix them.

Try it

Free resources for this checkpoint

Practice next

Question Bank filter: Math → Algebra → Systems of two linear equations and Linear inequalities. Solve half by hand, half by graphing — you'll need both speeds.