Question

analyzing a linear combination of a system of equations
the linear combination method is applied to a system of equations:

1. $(4x + 10y = 12)\frac{1}{2}$

$(10x + 25y = 30) - \frac{1}{5}$

1. $2x + 5y = 6$

$-2x - 5y = -6$

1. $0 = 0$

what does $0 = 0$ indicate about the solutions of the system?
\\(\circ\\) there are 0 solutions to the system.
\\(\circ\\) the solution to the system is $(0, 0)$.
\\(\circ\\) there are solutions to the system at the $x$- and $y$-intercepts.
\\(\circ\\) there are infinitely many solutions to the system.

Explanation:

When solving a system of linear equations, arriving at a true statement like $0=0$ means the two original equations are equivalent (they represent the same line). Every point on that line is a solution, so there are infinitely many solutions. The other options are incorrect: 0 solutions would result in a false statement like $0=5$, $(0,0)$ is only one specific solution, and intercepts are just specific points not the full set of solutions.

Answer:

There are infinitely many solutions to the system.