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.
False.
Step 1: Write equations in standard form.
First equation: \(x - 2y = 8\) ⇒ \(x - 2y - 8 = 0\).
Second equation: \(5x - 10y = c\) ⇒ \(5x - 10y - c = 0\).
So, we have:
\(a_1 = 1,\; b_1 = -2,\; c_1 = -8\)
\(a_2 = 5,\; b_2 = -10,\; c_2 = -c\)
Step 2: Recall the condition for unique solution.
Two linear equations have a unique solution if:
\(\dfrac{a_1}{a_2} \ne \dfrac{b_1}{b_2}\).
Step 3: Compare ratios of coefficients.
\(\dfrac{a_1}{a_2} = \dfrac{1}{5}\).
\(\dfrac{b_1}{b_2} = \dfrac{-2}{-10} = \dfrac{1}{5}\).
The two ratios are equal.
Thus, the lines are either coincident or parallel.
Step 4: Check constants ratio.
\(\dfrac{c_1}{c_2} = \dfrac{-8}{-c} = \dfrac{8}{c}\).
• If \(\dfrac{8}{c} = \dfrac{1}{5}\), then \(c = 40\).
In this case, all three ratios are equal ⇒ the lines are coincident ⇒ infinitely many solutions.
• If \(c \ne 40\), then \(\dfrac{c_1}{c_2} \ne \dfrac{a_1}{a_2}\).
So the lines are parallel and distinct ⇒ no solution.
Step 5: Conclusion.
In neither case do we get a unique solution.
Hence, the given statement “For all real values of \(c\), the pair of equations have a unique solution” is false.