For which values of p and q, will the following pair of linear equations have infinitely many solutions? 4x + 5y = 2; (2p + 7q) x + (p + 8q) y = 2q - p + 1

Solution:

Given, the pair of linear equations is

4x + 5y = 2

(2p + 7q)x + (p + 8q)y = 2q - p + 1

We have to find the values of p and q for which the linear pair of equations will have infinitely many solutions.

We know that,

For a pair of linear equations in two variables be a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0,

If \(\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}=\frac{c_{1}}{c_{2}}\), then

i) the pair of linear equations is dependent and consistent

ii) the graph will be a pair of coincident lines. Each point on the lines will be a solution and so the pair of equations will have infinitely many solutions.

Here, a₁ = 4, b₁ = 5, c₁ = 2

a₂ = (2p + 7q), b₂ = (p + 8q), c₂ = 2q - p + 1

So, a₁/a₂ = 4/(2p + 7q)

b₁/b₂ = 5/(p + 8q)

c₁/c₂ = 2/(2q - p + 1)

So, 4/(2p + 7q) = 5/(p + 8q) = 2/(2q - p + 1)

Case 1) 4/(2p + 7q) = 5/(p + 8q)

On cross multiplication,

4(p + 8q) = 5(2p + 7q)

4p + 32 q = 10p + 35q

4p - 10p = 35q - 32q

-6p = 3q

Dividing by 3 on both sides,

q = -2p ------------ (1)

Case 2) 4/(2p + 7q) = 2/(2q - p + 1)

On cross multiplication,

4(2q - p + 1) = 2(2p + 7q)

8q - 4p + 4 = 4p + 14q

By grouping,

8q - 14q - 4p - 4p + 4 = 0

-6q - 8p = -4

Now, 8p + 6q = 4 ------------------ (2)

Substitute (1) in (2),

8p + 6(-2p) = 4

8p - 12p = 4

-4p = 4

p = -4/4

p = -1

Put p = -1 in (1),

q = -2(-1)

q = 2

Therefore, the values of p and q are -1 and 2

✦ Try This: For which values of p and q, will the following pair of linear equations have infinitely many solutions? 5x + 4y = 12; (p + 7q) x + (3p + 2q) y = 2q - p + 1

☛ Also Check: NCERT Solutions for Class 10 Maths Chapter 3

NCERT Exemplar Class 10 Maths Exercise 3.3 Sample Problem 1

For which values of p and q, will the following pair of linear equations have infinitely many solutions? 4x + 5y = 2; (2p + 7q) x + (p + 8q) y = 2q - p + 1

Summary:

For values of p = -1 and q =2, the pair of linear equations 4x + 5y = 2; (2p + 7q) x + (p + 8q) y = 2q - p + 1 has infinitely many solutions.

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