Question

choose the correct word to make a true statement about the system of equations.
$y = \frac{1}{5}x - \frac{2}{5}$
$x - 5y = 2$
the system has:
show your work here
\\(\circ\\) no solution \\(\circ\\) infinitely many solutions
\\(\circ\\) two solutions \\(\circ\\) one solution

Explanation:

Step1: Rewrite the second equation

We have the second equation \(x - 5y = 2\). Let's solve it for \(y\) to compare with the first equation. Subtract \(x\) from both sides: \(-5y=-x + 2\). Then divide both sides by \(-5\): \(y=\frac{1}{5}x-\frac{2}{5}\).

Step2: Compare the two equations

The first equation is \(y=\frac{1}{5}x-\frac{2}{5}\) and the second equation, after rearrangement, is also \(y=\frac{1}{5}x-\frac{2}{5}\). This means the two equations represent the same line.

Step3: Determine the number of solutions

If two equations represent the same line, then every point on the line is a solution to the system. So the system has infinitely many solutions.

Answer:

infinitely many solutions (the option: infinitely many solutions)