Question

the system of equations is solved using the linear combination method.
$\frac{1}{2}x + 4y=8$ $\to - 2(\frac{1}{2}x + 4y = 8)\to -x - 8y=-16$
$3x + 24y = 12\to\frac{1}{3}(3x + 24y = 12)\to x + 8y = 4$
$0=-12$
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, when after using a method like linear combination, we get a contradiction such as \(0=-12\), it means the system has no solutions. This is because the two equations in the system represent parallel lines. Parallel lines never intersect, and the solution of a system of linear equations is the point of intersection of the lines represented by the equations.

Answer:

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