Question

how many solutions does the system have?
\

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

choose 1 answer:
a exactly one solution
b no solutions
c infinitely many solutions

Explanation:

Step1: Analyze the system of equations

The system is \(

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

\). Let's denote the left - hand side of both equations as \(L = 5x - y\).

Step2: Compare the two equations

For the first equation, \(L=2\), and for the second equation, \(L = - 2\). But a single expression \(5x - y\) cannot be equal to two different values (\(2\) and \(-2\)) at the same time. Geometrically, these are two parallel lines (since they have the same slope, the slope of \(y = 5x-2\) and \(y = 5x + 2\) is \(5\)) and parallel lines that are not coincident (because the \(y\) - intercepts are different, \(-2\) and \(2\)) do not intersect. So, there are no solutions for this system of equations.

Answer:

B. No solutions