Question

the system of equations shown is solved using the linear combination method
6x - 5y = -8
-24x + 20y = 32
what does 0 = 0 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 statement like \(0 = 0\), it means that the two equations are equivalent. In other words, they represent the same line. When two equations represent the same line, every point on that line is a solution to the system. So, there are infinitely many solutions.

• If the equations represented parallel lines, we would get a contradiction like \(0= \text{non - zero number}\) (no solutions).
• If the equations were the same line, we get an identity (\(0 = 0\)) which implies infinitely many solutions.

Answer:

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