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

To demonstrate that the ratio can be expressed in the required form, we start with the initial ratio:

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

Cross-multiplying gives:

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

Expanding both sides results in:

$y + x = ky - kx$

Next, we rearrange the equation to isolate terms involving $y$:

$y - ky = -kx - x$

Factoring out $y$ on the left side leads to:

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

Finally, dividing both sides by $(1 - k)$ yields:

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

To simplify this further, we multiply the numerator and the denominator by -1, obtaining:

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

Thus, we successfully show the equivalency as required.