Question

The number of solutions of the pair of linear equations x+2y=8 and 2x+4y-16 are

• A. 0
• B. 1
• C. 2
• D. infinitely many

Answer 1

The Correct Option is D

\[ x + 2y = 8 \quad \text{and} \quad 2x + 4y = 16. \]

Step 1: Analyze the system of equations.

The first equation is:

\[ x + 2y = 8. \]

The second equation is:

\[ 2x + 4y = 16. \]

Step 2: Check if the equations are consistent or dependent.

Multiply the first equation by 2:

\[ 2(x + 2y) = 2(8) \implies 2x + 4y = 16. \]

This shows that the second equation is simply a multiple of the first equation. Hence, the two equations represent the same line.

Step 3: Determine the number of solutions.

Since the two equations represent the same line, they have infinitely many solutions (every point on the line satisfies both equations).

Final Answer: The number of solutions is \( \mathbf{\text{infinitely many}} \), which corresponds to option \( \mathbf{(4)} \).