Grade 10 Equations and Inequalities

Grade 10 equations and inequalities: linear, quadratic by factorising, literal equations, simultaneous equations and linear inequalities, with worked…

Question 1 of a Grade 10 Paper 1 is almost always equations. You solve linear equations, quadratic equations by factorising, literal equations (making a variable the subject), two linear equations simultaneously, simple exponential equations, and linear inequalities, showing the solution on a number line.

The quadratic formula is not part of Grade 10: every quadratic you meet will factorise. Grade 11 adds the formula, completing the square and quadratic inequalities, so a strong Grade 10 factorising habit pays off immediately.

Key concepts

  • Linear equations, including equations with fractions
  • Quadratic equations by factorising and the zero-product rule
  • Literal equations: changing the subject of a formula
  • Simultaneous linear equations by substitution or elimination
  • Linear inequalities and their solutions on a number line
  • Interval notation and set-builder notation

Worked examples

Solve: x² − 5x + 6 = 0

Factorise: (x − 2)(x − 3) = 0. By the zero-product rule, x = 2 or x = 3.

Solve simultaneously: x + y = 7 and 2x − y = 5

Add the equations to eliminate y: 3x = 12, so x = 4. Substitute into x + y = 7: y = 3.

Solve and represent on a number line: 3 − 2x > 7

Subtract 3: −2x > 4. Divide by −2 and flip the sign: x < −2. Draw an open dot at −2 with an arrow to the left.

Make n the subject of A = P(1 + in).

Divide both sides by P: A/P = 1 + in. Subtract 1: A/P − 1 = in. Divide by i: n = (A/P − 1)/i, which is the same as n = (A − P)/(Pi).

Common mistakes

  • Dividing both sides of a quadratic by x and losing the solution x = 0.
  • Forgetting to flip the inequality sign when multiplying or dividing by a negative.
  • Setting each factor equal to the right-hand side when it is not zero, e.g. (x − 1)(x + 2) = 4.
  • Making a slip with a negative when substituting back in simultaneous equations.

Exam tips

  • Make one side zero before you factorise a quadratic.
  • Use an open dot for < and > and a closed dot for ≤ and ≥ on number lines.
  • Check simultaneous solutions in both original equations.

Frequently asked questions

Do I need the quadratic formula in Grade 10?

No. CAPS introduces the quadratic formula in Grade 11. Grade 10 quadratics are solved by factorising.

What is a literal equation?

An equation with several letters, such as A = P(1 + in). Solving it means making one letter the subject, e.g. n = (A − P)/(Pi).

How do I choose between substitution and elimination?

If one equation already has a variable on its own (y = 2x + 1), substitute. If a variable has the same or opposite coefficients in both equations, eliminate by adding or subtracting.

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