Question

the system of equations is solved using the linear combination method.
\\( \frac { 1 } { 2 } x + 4 y = 8 \
ightarrow - 2 \left( \frac { 1 } { 2 } x + 4 y = 8 \
ight) \
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 the linear - combination method (or any other method for solving systems of equations) we get a statement like \(0=-12\) (a false statement), it means that the two equations have no common solution.

For a system of two linear equations \(a_1x + b_1y=c_1\) and \(a_2x + b_2y = c_2\), if the slopes are equal (\(-\frac{a_1}{b_1}=-\frac{a_2}{b_2}\)) and the \(y\) - intercepts are different (\(\frac{c_1}{b_1}
eq\frac{c_2}{b_2}\)), the lines are parallel. Parallel lines do not intersect, so there is no solution to the system of equations.

If the equations represented the same line, we would get a statement like \(0 = 0\) (after simplification). If there were infinitely many solutions, the two equations would be equivalent (same line).

Answer:

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