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2x + 3y - 5 = 0 and px - 6y - 8 = 0, if the pair of equations has a unique solution. Find the value(s) of p for the pair of equations

Solution:

Given, the pair of linear equations are

2x + 3y - 5 = 0

Px - 6y - 8 = 0

We have to determine the value of λ for which the pair of linear equations have a unique solution.

We know that,

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

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

i) The pair of linear equations is consistent

ii) The graph will be a pair of lines intersecting at a unique point, which is the solution of the pair of equations.

Here, a₁ = 2, b₁ = 3, c₁ = -5

a₂ = p, b₂ = -6, c₂ = -8

So, a₁/a₂ = 2/p

b₁/b₂ = 3/-6 = -1/2

c₁/c₂ = -5/-8 = 5/8

For unique solution,

\(\frac{a_{1}}{a_{2}}\neq \frac{b_{1}}{b_{2}}\)

So, 2/p ≠ -1/2

2(2) ≠ -p

p ≠ -4

So, all real values of p except -4.

Therefore, for all real values of p except p = -4, the pair of linear equations have a unique solution.

✦ Try This: For which value(s) of λ, do the pair of linear equations λx + 6y = 2λ and 8x + λy = 21 have a unique solution

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

NCERT Exemplar Class 10 Maths Exercise 3.3 Problem 4 (iv)

Summary:

For all real values of p except λ = -4, the pair of linear equations 2x + 3y - 5 = 0 and px - 6y - 8 = 0 has a unique solution.

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