Question

If $am=bl$, then find whether the pair of linear equation $ax+by=c$ and $lx+my=n$ has no solution, a unique solution or infinitely many solutions?

Answer 1

Step 1 : Given data

$$\begin{gathered} am=bl \\ \frac{a}{l}=\frac{b}{m}.....\left(i\right) \\ ax+by=c......\left(ii\right) \\ lx+my=n.......\left(iii\right) \end{gathered}$$

Step 2: Compare with Standard form

After comparing the given equation with the standard form of linear equation. we get,

$$\begin{gathered} a_1=a,b_1=b,c_1=-c \\ {a}_2=l,b_2=m,c_2=-n \end{gathered}$$

Now, from $\left(i\right),\left(ii\right)$ and $\left(iii\right)$

$$\begin{gathered} \frac{a_1}{a_2}=\frac{a}{l} \\ \frac{b_1}{b_2}=\frac{b}{m} \\ \frac{c_1}{c_2}=\frac{-c}{-n}=\frac{c}{n} \\ \therefore \frac{a}{l}=\frac{b}{m}\ne \frac{c}{n}(U\sin g(i\left)\right) \end{gathered}$$

Hence, the given pair of equations has no solution.