Question

Find the values of p and q for which the following system of linear equations has infinite number of solutions : 2x + 3y = 1, (p + q)x + (2p – q)y = 21

Answer 1

The given equations are : 

2x + 3y –1 = 0 and (p + q)x + (2p – q)y – 21 = 0

Here, a₁ = 2, b₁ = 3, c₁ = –1 and a₂ = p + q, b₂ = 2p – q, c₂ = – 21 

For infinite solutions,

\(\frac{a_1}{a_2}\) = \(\frac{b_1}{b_2}\)= \(\frac{c_1}{c_2}\) \(\Rightarrow\) \(\frac{2}{p+q}\) = \(\frac{3}{2p-q}\) = \(\frac{-1}{-21}\) \(\Rightarrow\) \(\frac{2}{p+q}\) = \(\frac{1}{21}\) and \(\frac{3}{2p-q}\) = \(\frac{1}{21}\)

\(\Rightarrow\) p + q = 42

Adding (1) and (2), we get

3p = 105 \(\Rightarrow\) p = 35

Putting p = 35 in (1), we get

35 + q = 42 \(\Rightarrow\) q = 42 - 35 = 7

\(\therefore\) p = 35, q = 7