Solving for the Ratio of x/y

The problem involves two distinct non-zero real variables, $x$ and $y$. We are given that $(x + y)$ is proportional to $(x - y)$. This proportionality implies the existence of a constant, let's call it $k$, such that:

• $(x + y) = k(x - y)$

Our goal is to determine the nature of the ratio $\frac{x}{y}$.

Deriving the Constant Ratio

To find the value of $\frac{x}{y}$, we need to manipulate the given proportionality equation:

1. Start with the proportionality equation: $x + y = k(x - y)$
2. Distribute the constant $k$ on the right side: $x + y = kx - ky$
3. Rearrange the terms to group $x$ terms together and $y$ terms together. Move $y$ terms to the left and $x$ terms to the right: $y + ky = kx - x$
4. Factor out $y$ from the left side and $x$ from the right side: $y(1 + k) = x(k - 1)$
5. To find the ratio $\frac{x}{y}$, divide both sides by $y$ (since $y \neq 0$) and by $(k - 1)$. Note that $k$ cannot be $1$, because if $k=1$, the equation becomes $y(2) = x(0)$, which implies $2y=0$, meaning $y=0$. However, $y$ is given as a non-zero variable. Therefore, $k \neq 1$, and $(k-1)$ is non-zero. $\frac{x}{y} = \frac{1 + k}{k - 1}$

Since $k$ represents a constant of proportionality, the expression $\frac{1 + k}{k - 1}$ will always yield a specific numerical value. This value does not depend on the specific values of $x$ or $y$, as long as they satisfy the initial condition. Therefore, the ratio $\frac{x}{y}$ is a constant.