Question

the system of equations shown is solved using the linear combination method
$6x - 5y = -8 \quad \
ightarrow \quad 6x - 5y = -8 \quad \
ightarrow \quad 6x - 5y = -8$
$-24x + 20y = 32 \quad \
ightarrow \quad \frac{1}{4}(-24x + 20y = 32) \quad \
ightarrow \quad -6x + 5y = 8$
\underline{$6x - 5y = -8$}
\underline{$-6x + 5y = 8$}
$0 = 0$
what does $0 = 0$ 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 the same line.

Explanation:

1. Analyze the equations: The second equation \(-24x + 20y = 32\) is multiplied by \(\frac{1}{4}\) to get \(-6x + 5y = 8\), which is \(-(6x - 5y) = 8\), and the first equation is \(6x - 5y=-8\), so \(-(6x - 5y)=8\) is equivalent to \(6x - 5y=-8\). So the two equations are the same line.
2. Recall the concept of solutions of a system of linear equations: If two equations represent the same line, then every point on the line is a solution, so there are infinitely many solutions. If \(0 = 0\) occurs during elimination, it means the equations are dependent (same line) and have infinitely many solutions. Options saying no solutions are wrong. The correct reasoning is that same line implies infinitely many solutions.

Answer:

There are infinitely many solutions to the system because the equations represent the same line.