Question

Find the value of a and b for which the following system of linear equations has infinite

number of solutions:

2x – 3y = 7 and ( a + b )x – ( a + b – 3 )y = 4a + b​

Answer 1

2x - 3y = 7 and (a + b)x - (a + b -3)y = 4a + b

On comparing both linear equations with standard form ( i.e., ax + by + c ) of linear equation, we get

a1 = 2 , b1 = -3 , c1 = 7

a2 = (a + b) , b2 = -(a + b - 3) , c2 = 4a + b

For Infinitely many solutions:

$</p><p>\sf \qquad \dfrac{a_{1}}{a_{2}} = \dfrac{ b_{1} }{ b_{2} } = \dfrac{c_{1}}{c_{2}} \\ \\ \sf \rightarrow \quad \dfrac{a_{1}}{a_{2}} = \dfrac{ b_{1} }{ b_{2} } \quad ; \quad \dfrac{ b_{1} }{ b_{2} } = \dfrac{c_{1}}{c_{2}} \\ \\</p><p>$

$</p><p> \sf \rightarrow \quad \frac{2}{a + b} = \frac{ \cancel{-} \: 3}{ \cancel{-} (a + b - 3)} \quad and \quad \frac{ \cancel{-} \: 3}{ \cancel{-} (a + b - 3)} = \frac{7}{4a + b} \\ \\ \sf \rightarrow \quad 2a + 2b - 6 = 3a + 3b \quad and \quad 12a + 3b = 7a + 7b - 21 \\ \\ \sf \rightarrow \quad a + b = - 6 \quad ...(1) \quad and \quad 5a - 4b = - 21 \quad ...(2) \\ \\</p><p>$

Multiply (1) by 5 , we get

☛ 5a + 5b = -30 ...(3)

Subtract (3) from (2)

☛ 5a - 4b - 5a - 5b = -21 + 30

☛ -9b = 9

☛ b = -1

Put value of b in (1)

☛ a + b = -6

☛ a - 1 = -6

☛ a = -5

Hence, value of a is -5 and value of b is -1