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

Next, we substitute this expression for y into the second equation:

y^2 + x^2 = 10 becomes (x - 2)^2 + x^2 = 10.

Now, we expand (x - 2)^2:

$(x - 2)^2 = x^2 - 4x + 4.$
Thus, the equation becomes:

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

Combining like terms gives:

$2x^2 - 4x + 4 = 10$. Subtracting 10 from both sides results in:

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

We can simplify by dividing the entire equation by 2:

$x^2 - 2x - 3 = 0.$ Factoring this, we get:

$(x - 3)(x + 1) = 0.$
Thus, the solutions for x are:

$x = 3$ or $x = -1.$

We will now find the corresponding y values for each x:

1. For $x = 3$:
   $y = 3 - 2 = 1.$ So, one solution is $(3, 1)$.

2. For $x = -1$:
   $y = -1 - 2 = -3.$ So, another solution is $(-1, -3).$

The solutions to the simultaneous equations are:

1. $(3, 1)$
2. $(-1, -3)$.