Question

Question 5
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 1

The given pair of linear equations are
$x–2y–8=0and5x–10y–c=0$
Here, $a_1=1b_1=-2c_1=-8a_2=5,b_2=-10,c_2=-c$

Now, $\frac{a_1}{a_2}=\frac{1}{5}$

and, $\frac{b_1}{b_2}=\frac{-2}{-10}=\frac{1}{5}$

Thus, $\frac{a_1}{a_2}=\frac{b_1}{b_2}$

Thus, the pair of equations represents either parallel or coincident lines

The pair of equations have either no solutions at all or infinitely many solutions.

Thus, no matter what the value of c is, the pair of equations will not have a unique solution.

Hence, the given statement is not true.