Solve the simultaneous equations y = x - 2, y^2 + x^2 = 10. - Edexcel - A-Level Maths Pure - Question 6 - 2007 - Paper 2

Now, let's expand $(x - 2)^2$:

$x^2 - 4x + 4 + x^2 = 10$

Combining like terms gives us:

$2x^2 - 4x + 4 = 10$

Rearranging this equation leads to:

$2x^2 - 4x + 4 - 10 = 0$

Simplifying gives us:

$2x^2 - 4x - 6 = 0$

We can divide the entire equation by 2:

$x^2 - 2x - 3 = 0$

Now, we can factor the equation:

$(x - 3)(x + 1) = 0$

Setting each factor to zero gives us:

$x - 3 = 0 \Rightarrow x = 3$ $x + 1 = 0 \Rightarrow x = -1$

Now, we substitute these values of x back into the first equation:

For $x = 3$: $y = 3 - 2 = 1$

For $x = -1$: $y = -1 - 2 = -3$

The solutions to the simultaneous equations are:

$(x, y) = (3, 1) \quad \text{and} \quad (x, y) = (-1, -3)$