Simultaneous equations and inequalities

Maths · Algebra. The idea in plain words, the rules to remember, the mistake to avoid and a worked example, written and checked by Mega Lecture.

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The idea

Simultaneous equations are two equations that are true at the same time, so one pair of values (x, y) fits both. With two linear equations, make the number in front of x or y match, then add or subtract the equations to remove that letter. With one linear and one quadratic equation, use substitution: put the linear one into the quadratic one. You usually get two pairs of answers.

Word problems work the same way once you turn the sentences into equations.

An inequality has a range of answers, not one value. Solve it like an equation, with one extra rule about negative numbers.

Rules to remember

Elimination: same signs subtract, different signs add.
Linear with quadratic: substitute, solve the quadratic, then find the other letter for each solution.
Multiplying or dividing an inequality by a negative number reverses the sign.
Number line: open circle for < or >, filled circle for ≤ or ≥.
Set notation: write the answer as {x : x > 3}.
Quadratic with roots p < q and positive x² term: "< 0" gives p < x < q; "> 0" gives x < p or x > q.
Graphs: dashed line for < or >, solid line for ≤ or ≥; test a point to find the correct side.

Common mistake

Forgetting to reverse the inequality sign when dividing by a negative number. For example −3x < 12 gives x > −4, not x < −4.

Worked example

Solve the simultaneous equations 4x + 3y = 5 and 2x − y = 5.

Multiply the second equation by 3: 6x − 3y = 15.
The y terms have different signs, so add: 4x + 6x = 5 + 15, so 10x = 20 and x = 2.
Put x = 2 into 2x − y = 5: 4 − y = 5, so y = −1.
Check in the first equation: 4(2) + 3(−1) = 8 − 3 = 5.

Answer: x = 2, y = −1

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