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Simultaneous Equations

Algebra and Sequences

Always substitute

  • Where one equation contains a squared term, elimination will not work and substitution is used instead
  • Make an unknown the subject of the linear equation, then substitute into the non-linear one
  • The result is a quadratic, so bring everything to one side to equal zero and factorise
The curve y = x² + 3x + 1 and the line y = 2x + 1 on the same axes, meeting at two points, (−1, −1) and (0, 1), which is why a linear and a non-linear pair has two solutions rather than one
Source: Quadratic simultaneous equations by Save My Exams
  • Two solutions for one unknown give two matching values of the other, so answers come in pairs
  • A quadratic with a repeated root gives a single solution pair, and graphically that is a line touching the curve rather than cutting it
  • A quadratic with no real solutions means the line and the curve never meet, so the pair has no solution at all
  • Powers no higher than two appear in this specification
Exam tip

There are usually two pairs of answers, and each x must be paired with its own y. Substitute each x back into the linear equation, which is quicker and less error-prone, and set the answers out as pairs rather than as two separate lists.

Worked example

A linear and a quadratic equation

Solve y = x² − 4 and y = 3x simultaneously.

Solution:

  • Elimination does not work on a non-linear pair, so always substitute
  • Both right-hand sides are equal to y, so they are equal to each other: x² − 4 = 3x
  • One unknown is now gone, leaving a quadratic to solve
  • Bring every term to one side so it equals zero, which is the only form that can be factorised usefully: x² − 3x − 4 = 0
  • Two numbers multiplying to −4 and adding to −3 are −4 and 1
  • Factorise: (x − 4)(x + 1) = 0
  • A product is zero only if a bracket is zero, so x = 4 or x = −1
  • Substitute each value back to get its partner, using the simpler equation y = 3x
  • The solutions are x = 4, y = 12 and x = −1, y = −3
  • Expect two solution pairs here and give both, keeping each x with its own y
  • Check the first in the other equation: 4² − 4 = 12
  • Graphically these solutions are the points where the line and the curve cross, which is why there can be two of them