Question

Solve the following system of linear equations for$xandy$

$\frac{x}{a}+\frac{y}{b}=2;bx-ay=a+b$

Answer 1

Step 1: Simplifying the equation by taking the LCM

Given,

$$\begin{gathered} \frac{x}{a}+\frac{y}{b}=2 \\ \Rightarrow \frac{bx+ay}{ab}=2[takingabasLCM] \\ \Rightarrow bx+ay=2ab \\ \Rightarrow bx+ay=2ab..........\left(1\right) \\ bx-ay=a+b..........\left(2\right) \end{gathered}$$

Step 2: Applying elimination method

$\left(1\right)+\left(2\right)$

$$\begin{gathered} \Rightarrow 2bx=a+b+2ab \\ \Rightarrow x=\frac{a+b+2ab}{2b} \end{gathered}$$

Step 3:To find the value of $y$

Substituting the value of $x=\frac{a+b+2ab}{2b}$ in equation $\left(1\right)$

$$\begin{gathered} b\left[\frac{a+b+2ab}{2b}\right]+ay=2ab \\ \Rightarrow \frac{a+b+2ab}{2}-2ab=-ay \\ \Rightarrow \frac{a+b+2ab-4ab}{2a}=-y \\ \Rightarrow \frac{2ab-a-b}{2a}=y \end{gathered}$$

Hence the values of $x=\frac{a+b+2ab}{2b}andy=\frac{2ab-a-b}{2a}$