Question

\

$$\begin{cases} x - y = 7 \\\\ x - y = -4 \\end{cases}$$

Explanation:

Step1: Analyze the system of equations

The given system is \(

$$\begin{cases}x - y = 7\\x - y = - 4\end{cases}$$

\). Let's rewrite both equations in slope - intercept form (\(y=mx + b\), where \(m\) is the slope and \(b\) is the y - intercept).
For the first equation \(x - y=7\), we can solve for \(y\):
\(y=x - 7\), so the slope \(m_1 = 1\) and the y - intercept \(b_1=-7\).
For the second equation \(x - y=-4\), we solve for \(y\):
\(y=x + 4\), so the slope \(m_2 = 1\) and the y - intercept \(b_2 = 4\).

Step2: Determine the nature of the lines

Since the slopes of the two lines (\(m_1=m_2 = 1\)) are equal and the y - intercepts (\(b_1=-7\) and \(b_2 = 4\)) are different, the two lines are parallel. Parallel lines never intersect, so the system of equations has no solution.

Step3: Graphical interpretation (optional but for verification)

If we were to graph \(y=x - 7\) and \(y=x + 4\), we would start by plotting the y - intercepts. For \(y=x - 7\), we plot the point \((0,-7)\) and then use the slope (rise 1, run 1) to draw the line. For \(y=x + 4\), we plot the point \((0,4)\) and then use the slope (rise 1, run 1) to draw the line. We would observe that the two lines are parallel and do not intersect.

Answer:

The system of equations \(

$$\begin{cases}x - y = 7\\x - y=-4\end{cases}$$

\) has no solution because the two lines represented by the equations are parallel (they have the same slope \(m = 1\) and different y - intercepts) and parallel lines do not intersect.