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- 3x + 5y = 7 and 2px - 3y = 1, if the lines represented by these equations are intersecting at a unique point. Find the value(s) of p for the pair of equations

Solution:

-3x + 5y = 7

2px - 3y = 1

We have to determine the value of p for which the lines represented by the pair of linear equations are intersecting at a unique point.

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 = -3, b₁ = 5, c₁ = 7

a₂ = 2p, b₂ = -3, c₂ = 1

So, a₁/a₂ = -3/2p

b₁/b₂ = 5/-3 = -5/3

c₁/c₂ = 7/1 = 7

For unique solution,

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

So, -3/2p ≠ -5/3

3/2p ≠ 5/3

3(3) ≠ 5(2p)

9 ≠ 10p

p ≠ 9/10

So, all real values of p except 9/10.

Therefore, for all real values of p except p ≠ 9/10, the pair of linear equations has a unique solution.

✦ Try This: For which value(s) of λ, do the pair of linear equations λx + 6y = 32λ and 6x + λy = 31 have a unique solution

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

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

- 3x + 5y = 7 and 2px - 3y = 1, if the lines represented by these equations are intersecting at a unique point. Find the value(s) of p for the pair of equations

Summary:

For all real values of p except p ≠ 9/10, the lines represented by the pair of linear equations -3x + 5y = 7 and 2px - 3y = 1 are intersecting at a unique point.

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