Question

The number of solutions of the pair of linear equations \( x - y = 8 \) and \( 3x - 3y = 16 \) will be:

• A. Infinite
• B. None
• C. Only one
• D. Two

Hint:

For two linear equations, if \( \frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2} \), then the lines are parallel and have no solution.

Answer 1

The Correct Option is B

Step 1: Express the equations in standard form.
Equation (1): \( x - y - 8 = 0 \) → \( a_1 = 1, b_1 = -1, c_1 = -8 \)
Equation (2): \( 3x - 3y - 16 = 0 \) → \( a_2 = 3, b_2 = -3, c_2 = -16 \)

Step 2: Compare ratios.
\[ \frac{a_1}{a_2} = \frac{1}{3}, \quad \frac{b_1}{b_2} = \frac{-1}{-3} = \frac{1}{3}, \quad \frac{c_1}{c_2} = \frac{-8}{-16} = \frac{1}{2} \]
Step 3: Analyze the ratios.
We have \[ \frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2} \] This condition means the pair of lines are parallel and distinct.
Step 4: Conclusion.
Hence, there is no solution to this pair of linear equations.