Question

Solve the following system of linear equations for $xandy$

$ax+by=a-b;bx-ay=a+b$

Answer 1

Step 1:Making the equations with $y$ coefficient as same

Given,

$$\begin{gathered} ax+by=a-b..........\left(1\right) \\ bx-ay=a+b..........\left(2\right) \end{gathered}$$

$\left(1\right)\times a\Rightarrow a^{2}x+aby=a^2-ab.......\left(3\right)$Making ‘$y$’ coefficient as same

$\left(2\right)\times b\Rightarrow b^{2}x-aby=ab+b^2.......\left(4\right)$ Making ‘$y$’ coefficient as same

Step 2:Applying elimination method to find the value of $x$

$\left(3\right)+\left(4\right)$

$$\begin{gathered} \Rightarrow a^{2}x+b^{2}x=a^2+b^2 \\ \Rightarrow x(a^2+b^2)=a^2+b^2 \\ \Rightarrow x=\frac{a^2+b^2}{a^2+b^2} \\ \Rightarrow x=1 \end{gathered}$$

Step 3:find the value of $y$ by substitution methods

Substituting $x=1\text{in eq}\left(1\right)$

$$\begin{gathered} \left(1\right)\Rightarrow a+by=a-b \\ \Rightarrow by=a-b-a \\ \Rightarrow by=-b \\ \Rightarrow y=-1 \end{gathered}$$

Hence the values of $x=1\text{and}y=-1$