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 first equation

From \( x - 5y - 1 = 0 \), we can express \( x \) as \( x = 5y + 1 \).

Step2: Substitute \( x \) into the second equation

The second equation is \( -2 = 10y - 2x \). Substitute \( x = 5y + 1 \) into it:
\[

$$\begin{align*} -2&=10y - 2(5y + 1)\\ -2&=10y - 10y - 2\\ -2&=-2 \end{align*}$$

\]
This is a true statement, which means the two equations are equivalent (they represent the same line), so there are infinitely many solutions. The solution set is all the points that satisfy either of the original equations.

Answer:

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