Simultaneous Equations Linear/Quadratic
# Simultaneous Equations Linear/Quadratic

GCSE(H),

Simultaneous equations can also be solved when one of the equations is in a quadratic form. In this instance:

• Make the subject of the linear equation the same as the power term in the quadratic equation;

• Substitute into the quadratic equation;

• Solve the quadratic equation, using factorisation, completing the square, or the quadratic formula;

• Solve the second variable by substitution into the linear equation;

• Check answers by substitution.

## Examples

1. Solve `y=2x+10 text( and ) y=-x^2-4x+5`

Answer: `x=-1, y=8 text( and ) x=-5, y=0`

Substitute the linear equation into the quadratic equation using the variable `y`:

`2x+10=-x^2-4x+5`

`x^2+4x+2x+10-5=0`

`x^2+6x+5=0`

Factorise to `(x+1)(x+5)=0`

With `x=-1, y=8 text( and ) x=-5, y=0`

Check: 8 = -(-1)^{2} - 4(-1) + 5 (true) and 0 = -(-5)^{2} - 4(-5) + 5 (true)

2. Solve `x-2y=4 text( and ) 2y=x^2-8x+4`.

Answer: `x=1, y=-1frac(1)(2) text( and ) x=8, y=2`

Rearrange `x-2y=4 text( to ) 2y=x-4`

Substitute for the linear equation; `x-4=x^2-8x+4`

Rearrange and set one side to zero: `x^2-8x-x-4-4=0`

Factorise `x^2 - 9x - 8` to `0=(x-1)(x-8)`

Substitute into the linear equation to get `y = -1frac(1)(2), y=2`

Check: `-1frac(1)(2) = 0.5(1)^2 - 4(1) + 2` (true) and `(2) = 0.5(8)^2 - 4(8) + 2` (true)

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