Question

The pair of equations λx + 3y = 7, 2x + 6y = 14 will have infinitely many solutions for λ = ________.

Answer 1

If the system of equations $a_{1}x+b_{1}y+c_1=0\mathrm{and}{a}_{2}x+b_{2}y+c_2=0$ has infinitely many solutions, then $\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}$.

Since, the system of equations λx + 3y = 7, 2x + 6y = 14 has infinitely many solutions
Therefore,
$$\begin{gathered} \frac{\lambda }{2}=\frac{3}{6}=\frac{-7}{-14} \\ \Rightarrow \frac{\lambda }{2}=\frac{1}{2}=\frac{1}{2} \\ \Rightarrow \frac{\lambda }{2}=\frac{1}{2} \\ \Rightarrow \lambda =1 \end{gathered}$$

Hence, the pair of equations λx + 3y = 7, 2x + 6y = 14 will have infinitely many solutions for λ = 1 .