The ratio $(y + x):(y - x)$ is equivalent to $k:1$ - Edexcel - GCSE Maths - Question 14 - 2017 - Paper 1

To show that the ratios are equivalent, we start with the given equation:

$\frac{y + x}{y - x} = \frac{k}{1}$

Cross-multiplying gives:

$(y + x) \cdot 1 = k(y - x)$

Expanding both sides results in:

$y + x = ky - kx$

Now, rearranging the terms to isolate $y$:

$y - ky = -kx - x$

Factoring out $y$ from the left side:

$y(1 - k) = -x(k + 1)$

Dividing both sides by $(1 - k)$ (assuming $k \neq 1$):

$y = \frac{-x(k + 1)}{1 - k}$

Now, simplifying the right side:

$y = \frac{x(k + 1)}{k - 1}$

This shows that the expression for $y$ confirms the equation as required.