Simultaneous equations

Two linear equations

Use elimination (add or subtract to remove one letter) or substitution. Check your answer in both equations.

Linear and quadratic

1. Rearrange the linear equation to make x or y the subject.

2. Substitute into the quadratic.

3. Solve the quadratic.

4. Substitute each value back into the linear equation to find the partner value.

Give the answers as pairs.

What the quadratic tells you

The number of solutions is the number of intersection points. If the quadratic you get has b² - 4ac < 0 the graphs do not meet; = 0 means the line is a tangent; > 0 means two crossing points.

Worked example

Solve y = x + 1 and x² + y² = 25.

1. Substitute: x² + (x + 1)² = 25
2. 2x² + 2x - 24 = 0, so x² + x - 12 = 0
3. (x + 4)(x - 3) = 0, so x = 3 or x = -4
4. y = x + 1 gives y = 4 or y = -3

Answer: x = 3, y = 4 or x = -4, y = -3

Check yourself

1. Solve 3x + 2y = 12 and x - y = -1. Find x.
   Show answer

   2. From the second equation x = y - 1. Then 3(y - 1) + 2y = 12, so 5y = 15, y = 3 and x = 2.

2. Solve 3x + 2y = 12 and x - y = -1. Find y.
   Show answer

   3. 5y = 15, so y = 3.

3. y = 2x - 1 and y = x² - 4. Find the x-coordinates of the points where they meet.
   Show answer

   x = 3 or x = -1. x² - 4 = 2x - 1 gives x² - 2x - 3 = 0, so (x - 3)(x + 1) = 0.

4. Solve y = x + 3 and xy = 10. Give the possible values of x.
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   x = 2 or x = -5. x(x + 3) = 10 gives x² + 3x - 10 = 0, so (x + 5)(x - 2) = 0.

5. The line y = 2x + k is a tangent to the curve y = x² + 1. Find k.
   Show answer

   0. x² + 1 = 2x + k gives x² - 2x + (1 - k) = 0. A tangent means equal roots: 4 - 4(1 - k) = 0, so 4k = 0 and k = 0.

6. After substituting a line into a curve, you get a quadratic with b² - 4ac < 0. What does this tell you?
   • The line is a tangent to the curve
   • The line crosses the curve twice
   • The line is parallel to the x-axis
   • The line and the curve do not meet
   Show answer

   The line and the curve do not meet. No real roots means no points of intersection.

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Sources

• Pearson Edexcel International Advanced Level in Mathematics specification (issue 3)
• Pearson Edexcel IAL Pure Mathematics 1 scheme of work

Last updated 7 November 2026. Spotted a mistake? Tell us.