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 \underline { 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:

In a system of linear equations, if after using elimination (linear combination) we get a false statement like \(0=-12\), it means the system is inconsistent. For a system of two - variable linear equations \(a_1x + b_1y=c_1\) and \(a_2x + b_2y=c_2\), the equations can be written in slope - intercept form \(y=mx + b\) (where \(m\) is the slope).
The first equation \(\frac{1}{2}x + 4y=8\) can be rewritten as \(y=-\frac{1}{8}x + 2\) (slope \(m_1=-\frac{1}{8}\)), and the second equation \(3x + 24y=12\) can be rewritten as \(y=-\frac{1}{8}x+\frac{1}{2}\) (slope \(m_2 =-\frac{1}{8}\)). Since the slopes are equal (\(m_1 = m_2\)) and the y - intercepts (\(b_1 = 2\) and \(b_2=\frac{1}{2}\)) are different, the lines are parallel. Parallel lines do not intersect, and an inconsistent system (a false statement like \(0=-12\)) implies no solutions.

Answer:

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