Question

Given the proportions $\frac{x+y}{y}=\frac{r}{s}$and $\frac{x-y}{x+y}=\frac{s}{y}$, what can you conclude?

Short answer

The solution of the proportion is, $r=x-y$

Step-by-step solution

Step 1. Given information.

$\frac{x+y}{y}=\frac{r}{s}$ and $\frac{x-y}{x+y}=\frac{s}{y}$

Given expression,

$\frac{x+y}{y}=\frac{r}{s}$ and $\frac{x-y}{x+y}=\frac{s}{y}$

Step 2. Solve the first equation.

$\Rightarrow \frac{x+y}{y}=\frac{r}{s}$

$\Rightarrow yr=s(x+y)$

$\Rightarrow yr=sx+sy\to \text{(i)}$

Step 3. Solve the second equation.

$\Rightarrow \frac{x-y}{x+y}=\frac{s}{y}$

$\Rightarrow s(x+y)=y(x-y)$

$\Rightarrow sx+sy=yx-y^2$

$\Rightarrow sx+sy=yx-y^2\to \text{(ii)}$

From equation (i) and (ii),

$\Rightarrow yr=yx-y^2$

Eliminating y from the whole equation,

$\Rightarrow r=x-y$

Therefore, the solution of the proportion is, $r=x-y$.