For all real values of c, the pair of equations x - 2y = 8 and 5x - 10

1.	For all real values of c, the pair of equations x - 2y = 8 and 5x - 10y = c have a unique solution. Justify whether it is true or false.
Answer» The answer is False. The given pair of linear equations x - 2y - 8 = 0 and 5x - 10y - c = 0 On Comparing with ax + by + c = 0; Here, Here, a₁ = 1, b₁ = - 2, c₁ = - 8; And a₂ = 5, b₂ = - 10, c₂ = - c; a₁ /a₂ = 1/5 b₁ /b₂ = 1/5 c₁ /c₂ = 8/c But if c = 40 (real value), then the ratio c₁/c₂ becomes 1/5 and then the system of linear equations has an infinitely many solutions. Hence, at c = 40, the system of linear equations does not have a unique solution.

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

Answer»

The answer is False.

The given pair of linear equations

x - 2y - 8 = 0

and 5x - 10y - c = 0

On Comparing with ax + by + c = 0;

Here,

Here, a₁ = 1, b₁ = - 2, c₁ = - 8;

And a₂ = 5, b₂ = - 10, c₂ = - c;

a₁ /a₂ = 1/5

b₁ /b₂ = 1/5

c₁ /c₂ = 8/c

But if c = 40 (real value), then the ratio c₁/c₂ becomes 1/5 and then the system of linear equations has an infinitely many solutions.

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