Q) If the system of linear equations 2 x + 3 y = 7 and 2 a x + (a + b) y = 28 have an infinite number of solutions, then find the values of ‘a’ and ‘b’.

Ans:

Step 1: We know that the standard form of a linear equation is: a x + b y + c = 0

Given 1st linear equation is: 2 x + 3 y = 7

In standard form, it can be written as: 2 x + 3 y – 7 = 0

Comparing it with standard form, we get:

a₁ = 2; b₁ = 3; c₁ = – 7

Similarly, when we compare 2nd linear equation (given) with standard form of equation, we get:

2 a x + (a + b) y = 28

or 2 a x + (a + b) y – 28 = 0

Therefore,  a₂ = 2 a ; b₂ = (a + b) ; c₂ = – 28

Step 2: Next we know, that when a system of linear equations has infinite solutions, then

∴

∴

Step 3: solving 1st and 3rd equations, we get:

∴ a = 4

Next, we take 2nd and 3rd equations, we get:

∴ a + b = 12

∴ 4 + b = 12

∴ b = 12 – 4 = 8

Therefore, the values are a = 4 and b = 8.

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