Question

The pair of equations 2x + y = 5, 3x + 2y = 8 has

विकल्प

• a unique solution

• two solutions

• no solution

• infinitely many solutions

Answer 1

A unique solution

Explanation:

1. Identify coefficients

Compare the given linear equations with the standard forms a₁x + b₁y = c₁ and a₂x + b₂y = c₂:

First equation: 2x + y = 5

⇒ a₁ = 2, b₁ = 1, c₁ = 5

Second equation: 3x + 2y = 8

⇒ a₂ = 3, b₂ = 2, c₂ = 8

2. Compare ratios

Calculate and compare the ratios of the coefficients of x and y:

`(a_1)/(a_2) = 2/3`

`(b_1)/(b_2) = 1/2`

Since `2/3 ≠ 1/2`, we have the condition:

`(a_1)/(a_2) ≠ (b_1)/(b_2)`

This condition dictates that the two lines intersect at exactly one point, meaning the system has one unique solution.