0580

Forming and Solving Equations

Algebra and Sequences

Using the property of the shape

  • A shape supplies the equation through one of its known properties
  • The most useful are the perimeter as a sum of sides, the angle sum of a triangle being 180°, the angle sum of a quadrilateral being 360°, and the equal sides or angles of an isosceles triangle
  • For a polygon with n sides, the interior angles sum to 180(n − 2)
  • Mark the diagram with the expressions before forming the equation
Exam tip

Write the equation on its own line before solving it: it carries marks even when the solving goes wrong. On a shape, the property — an angle sum, a perimeter — is what supplies the equation, so name it before writing anything. Then answer what was asked and attach units, rather than stopping at x.

Worked example

Perimeter of a rectangle

A rectangle has length (2x + 1) cm and width (x − 3) cm. Its perimeter is 38 cm. Find the length and the width.

A rectangle with its longer horizontal side labelled (2x + 1) cm and its shorter vertical side labelled (x − 3) cm. The perimeter is not shown on the figure. The figure is marked NOT TO SCALE.

Solution:

  • The property of the shape supplies the equation, so name it before writing anything: a rectangle's perimeter is twice the length plus twice the width
  • 2((2x + 1) + (x − 3)) = 38
  • Simplify inside the bracket by collecting like terms: 2x + x = 3x, and 1 − 3 = −2
  • 2(3x − 2) = 38
  • Divide both sides by 2 rather than expanding, which keeps the numbers small: 3x − 2 = 19
  • Add 2: 3x = 21, so x = 7
  • x = 7 is not the answer — the question asks for two lengths
  • Length = 2 × 7 + 1 = 15 cm, width = 7 − 3 = 4 cm
  • Check against the perimeter given: 2(15 + 4) = 2 × 19 = 38 cm, and both sides are positive, which a length must be