Question

For all real values of c, the pair of equations x – 2y = 8, 5x – 10y = c have a unique solution. Justify whether it is true or false.

पर्याय

• True

• False

Answer 1

This statement is False.

Explanation:

The given pair of linear equations

x – 2y – 8 = 0

5x – 10y = c

Here, a₁ = 1, b₁ = – 2, c₁ = a – 8

a₂ = 5, b₂ = – 10, c₂ = – c

Now, ${\displaystyle \frac{a_1}{a_2}=\frac{1}{5},\frac{b_1}{b_2}=\frac{-2}{-10}=\frac{1}{5}}$

${\displaystyle \frac{c_1}{c_2}=\frac{-8}{-c}=\frac{8}{c}}$

But if c = 40 (real value),

Then the ratio ${\displaystyle \frac{c_1}{c_2}}$ becomes ${\displaystyle \frac{1}{5}}$ and then the system of linear equations has an infinitely many solutions.

Hence, ate = 40, the system of linear equations does not have a unique solution.