Question

For what value(s) of c, the pair of equations \(x - 2y = 8; 5x - 10y = c\) have infinite solution ?

• A. 16
• B. All real values
• C. 40
• D. All non-negative real values of c

Hint:

Notice that the second equation has its left-hand side coefficients (\(5x - 10y\)) exactly \(5\) times the first equation's coefficients (\(x - 2y\)).
For the equations to represent the same line (infinite solutions), the constant term must also be multiplied by the same factor of \(5\).
Thus, simply multiply \(8 \times 5 = 40\) to find the answer in seconds.

Answer 1

The Correct Option is C

Step 1: Notice a shortcut hiding in the numbers.
Look at the left hand sides of the two equations, $x - 2y$ and $5x - 10y$. The second is exactly 5 times the first, since $5x - 10y = 5(x - 2y)$.
Step 2: Use this scaling directly.
For the two lines to be the same line, which is what infinite solutions means, the right hand sides must scale by the same factor of 5. Since the first equation gives $x - 2y = 8$, the second must satisfy \[ c = 5 \times 8 \]
Step 3: Compute the value.
\[ c = 40 \]
Step 4: Sanity check.
With $c = 40$, the equation $5x - 10y = 40$ is just 5 times $x - 2y = 8$, so every point on one line lies on the other, giving infinitely many common solutions, exactly as required.
\[ \boxed{c = 40} \]