Question

solve the following system by graphing.\\(
\

$$\begin{cases} 2x - y = 8 \\\\ 8x - 4y = 8 \\end{cases}$$

\\)\\(\dots\\)use the graphing tool to graph the lines.\\(\text{click to enlarge graph}\\)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 \\(\square\\).\\((\text{simplify your answer. type an ordered pair.})\\)\\(\bigcirc\\) b. the solution set of the system is \\(\\{(x,y)|2x - y = 8\\}\\) or \\(\\{(x,y)|8x - 4y = 8\\}\\).\\(\bigcirc\\) c. the solution set is \\(\varnothing\\).

Explanation:

Step1: Rewrite equations in slope-intercept form

For \(2x - y = 8\), solve for \(y\):
\(y = 2x - 8\).

For \(8x - 4y = 8\), divide by 4: \(2x - y = 2\), then solve for \(y\):
\(y = 2x - 2\).

Step2: Analyze slopes and intercepts

Both lines have slope \(m = 2\) (same slope) but different y-intercepts (\(-8\) vs. \(-2\)). Parallel lines never intersect.

Answer:

C. The solution set is \(\varnothing\).