Question

the system of equations is solved using the linear combination method.
$\frac{1}{2}x + 4y = 8 \to -2\left(\frac{1}{2}x + 4y = 8\
ight) \to -x - 8y = -16$
$3x + 24y = 12 \to \frac{1}{3}(3x + 24y = 12) \to x + 8y = 4$
$\overline{0 = -12}$
what does $0 = -12$ mean regarding the solution to the system?
$\bigcirc$ there are no solutions to the system because the equations represent parallel lines.
$\bigcirc$ there are no solutions to the system because the equations represent the same line.
$\bigcirc$ there are infinitely many solutions to the system because the equations represent parallel lines.

Explanation:

When solving a system of linear equations, if we get a false statement like \(0 = -12\), it means the lines are parallel (no intersection, so no solution). The first option says this. The second is wrong (same line would give infinite solutions, not \(0=-12\)), third is wrong (parallel lines have no solutions, not infinite).

Answer:

A. There are no solutions to the system because the equations represent parallel lines.