Question

solve the system by the method of your choice.\

$$\begin{cases}x - 5y - 1 = 0\\\\-2 = 10y - 2x\\end{cases}$$

\select the correct choice below and, if necessary, fill in the answer box to complete your choice.\\(\bigcirc\\) a. there is exactly one solution. the solution set is \\{\\ \\}. (simplify your answer. type an ordered pair.)\\(\bigcirc\\) b. there are infinitely many solutions. the solution set is \\{\\(x,y\\)\mid (x - 5y - 1 = 0\\} or \\{\\(x,y\\)\mid (-2 = 10y - 2x\\}.\\(\bigcirc\\) c. the solution set is \\(\varnothing\\).

Explanation:

Step1: Rewrite the equations in standard form

First equation: \(x - 5y - 1 = 0\) can be rewritten as \(x - 5y = 1\).
Second equation: \(-2 = 10y - 2x\) can be rearranged to \(2x - 10y = 2\). Then divide this equation by 2, we get \(x - 5y = 1\).

Step2: Analyze the two equations

Now we have two equations:

1. \(x - 5y = 1\)
2. \(x - 5y = 1\)

These two equations are identical. This means that every point \((x,y)\) that satisfies the first equation will also satisfy the second equation. So there are infinitely many solutions, and the solution set is all the points on the line defined by either of the two equations.

Answer:

B. There are infinitely many solutions. The solution set is \(\{(x,y)\mid x - 5y - 1 = 0\}\) or \(\{(x,y)\mid - 2 = 10y - 2x\}\).