Equations and Inequalities: Linear Equations, Systems, Inequalities
What you must be able to do
Solve a linear equation by performing the same operation on both sides until the unknown stands alone. Clear denominators by multiplying by the least common multiple, expand brackets carefully, collect variables on one side and numbers on the other. A linear equation in one variable has one solution, no solution (a false statement such as 0 = 5), or infinitely many (0 = 0).
Solve systems of two linear equations by substitution or elimination. Elimination is usually faster when coefficients are close: multiply one or both equations to match a coefficient, then add or subtract.
Solve linear inequalities exactly as equations, with one extra rule: multiplying or dividing by a negative number reverses the inequality sign. Show the solution set on a number line, using an open circle for < or > and a filled circle for ≤ or ≥.
Traps that appear in questions
- Forgetting to reverse the sign when dividing by a negative: −2x > 6 gives x < −3.
- Multiplying only part of an equation when clearing denominators.
- Losing a minus sign when distributing over a bracket preceded by a minus.
- Accepting a solution without checking it in the original equation, especially after squaring or clearing denominators.
- In systems, adding equations but forgetting to add the right-hand sides.
Rule of thumb
Always substitute your answer back into the original equation or inequality; a ten-second check catches most sign errors.
Worked example
Solve 3(x − 1) < 5x + 7. 3x − 3 < 5x + 7 → −3 − 7 < 5x − 3x → −10 < 2x → x > −5.