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SQA National 5 Mathematics

Simultaneous equations

Solving two linear equations by elimination or substitution.

Formula referenceMixed revision

Formula reference

Check the National 5 rules and formulae linked to this topic.

Which method?

Match exam clues to a suitable method.

Before you start

  • Be confident solving linear equations.
  • Line up like terms carefully, especially x terms and y terms.
  • Remember that a solution must make both equations true.
National 5 Mathematics lesson

Explanation

Simultaneous equations are two equations that are true at the same time. At National 5, the usual task is to find the values of two unknowns, often x and y.

Elimination works by adding or subtracting the equations so that one unknown disappears. If the coefficients do not match, multiply one equation first.

Substitution works well when one equation already gives x or y on its own, for example y = x + 3. Substitute that expression into the other equation.

On a graph, the solution is the point where the two straight lines intersect. Algebra gives the exact coordinates of that intersection.

Key formulae and rules

  • Elimination: make one pair of coefficients equal, then add or subtract the equations.
  • Substitution: replace one unknown with an equivalent expression from the other equation.
  • Check: substitute x and y into both original equations.
  • Graph meaning: the solution is the intersection point of the two lines.

Watch out

Eliminating the wrong term after multiplying only one side of an equation.

Check

Compare your answer with the size you expected from the question.

Exam tip

Label the original equations and show the elimination or substitution step clearly. Always finish with both unknowns.

Calculator tip

Use brackets for fractions, powers and square roots, then round only at the final line.

Worked examples

Worked example 1

Add to eliminate y

Solve x + y = 7 and x − y = 1.

  1. Label the equations (1) and (2).
  2. Add them: (x + y) + (x − y) = 7 + 1
  3. This gives 2x = 8, so x = 4.
  4. Substitute into x + y = 7: 4 + y = 7

Answer: x = 4 and y = 3. Check: 4 − 3 = 1

Worked example 2

Subtract to eliminate y

Solve 2x + y = 11 and x + y = 7.

  1. Subtract the second equation from the first.
  2. (2x + y) − (x + y) = 11 − 7
  3. This gives x = 4.
  4. Substitute into x + y = 7

So: x = 4 and y = 3

Worked example 3

Multiply before eliminating

Solve 2x + y = 13 and x + 2y = 11.

  1. Multiply the first equation by 2: 4x + 2y = 26
  2. Subtract x + 2y = 11 from it
  3. 3x = 15, so x = 5
  4. Substitute into 2x + y = 13: 10 + y = 13

Answer: x = 5 and y = 3

Worked example 4

Substitution method

Solve y = x + 2 and 3x + y = 14.

  1. Substitute x + 2 for y in the second equation.
  2. 3x + (x + 2) = 14
  3. 4x + 2 = 14, so x = 3
  4. Then y = 3 + 2

So: x = 3 and y = 5

Watch out

  • Eliminating the wrong term after multiplying only one side of an equation.
  • Finding x but forgetting to substitute back to find y.
  • Checking the answer in only one of the two original equations.