Question

Solve for $xandy$

$\frac{13}{x+y}-\frac{56}{x-y}=21;\frac{11}{x+y}-\frac{23}{x-y}=14\frac{2}{7}$

Answer 1

Step 1 : Giving substitution

$$\begin{gathered} \frac{13}{x+y}-\frac{56}{x-y}=21 \\ \frac{11}{x+y}-\frac{23}{x-y}=\frac{100}{7} \end{gathered}$$

Let

$$\begin{gathered} \frac{1}{x+y}=Aand\frac{1}{x-y}=B \\ Therefore,13A-56B=21...................................\left(1\right) \\ 11A-23B=\frac{100}{7}..........................................\left(2\right) \\ \left(1\right)x23\Rightarrow 299A-1288B=483................................\left(3\right) \\ \left(2\right)x56\Rightarrow 616A-1288B=800................................\left(4\right) \end{gathered}$$

Step 2 : To find A using elimination method

$$\begin{gathered} equation\left(4\right)-\left(3\right)\Rightarrow 317A=317 \\ \Rightarrow A=1 \end{gathered}$$

Step 3 : To get B by substitution method

Substitute $A=1$in equation (1)

$$\begin{gathered} \left(1\right)\Rightarrow 13-56B=21 \\ \Rightarrow -56B=21-13 \\ \Rightarrow -56B=8 \\ \Rightarrow B=\frac{-8}{56}=\frac{-1}{7} \end{gathered}$$

Step 4: To get$\mathbf{}\mathit{x}\mathbf{}\mathit{a}\mathit{n}\mathit{d}\mathbf{}\mathit{y}\mathbf{}$by elimination method

We have,

$$\begin{gathered} A=\frac{1}{x+y} \\ \Rightarrow 1=\frac{1}{x+y} \\ \Rightarrow x+y=1...............................................................\left(5\right) \\ B=\frac{1}{x-y} \\ \Rightarrow \frac{-1}{7}=\frac{1}{x-y} \\ \Rightarrow -(x-y)=7 \\ \Rightarrow -x+y=7.............................................\left(6\right) \end{gathered}$$

We have

$$\begin{gathered} x+y=1 \\ -x+y=7 \\ equation\left(5\right)+\left(6\right) \\ 2y=8 \\ \Rightarrow y=4 \\ x=1-y=1-4=-3 \end{gathered}$$

Therefore the values of $x=-3andy=4$