Question

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

Explanation:

When using the linear combination method for a system of equations, if we end up with a false statement like \(0=-12\), it means the system is inconsistent. Inconsistent systems have no solutions. For a system of two - variable linear equations \(y = mx + b\) (where \(m\) is the slope and \(b\) is the \(y\) - intercept), if the slopes (\(m\) values) are equal (parallel lines) and the \(y\) - intercepts (\(b\) values) are different, the system is inconsistent. If the equations represented the same line (\(m\) and \(b\) are equal), we would have infinitely many solutions. Since \(0=-12\) is a false statement, the lines are parallel (same slope, different \(y\) - intercepts) and there are no solutions.

Answer:

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