Determine the solution to the system of inequalities. \[ \begin{array}{l} x-5 y \leq 5 \\ x+4 y \geq 4 \end{array} \]

Step 1 ：Rearrange the first inequality \(x - 5y \leq 5\) into slope-intercept form. Subtract x from both sides to get: \(-5y \leq 5 - x\). Then divide every term by -5 to solve for y. Remember that when you divide or multiply an inequality by a negative number, you must flip the inequality sign: \(y \geq (5 - x) / -5\) or \(y \geq -x/5 - 1\).

Step 2 ：Graph the first inequality. This is a line with a slope of -1/5 and a y-intercept of -1. Because the inequality is 'greater than or equal to', we will graph a solid line (indicating that points on the line are included in the solution) that slopes downward.

Step 3 ：Rearrange the second inequality \(x + 4y \geq 4\) into slope-intercept form. Subtract x from both sides to get: \(4y \geq 4 - x\). Then divide every term by 4 to solve for y: \(y \geq (4 - x) / 4\) or \(y \geq -x/4 + 1\).

Step 4 ：Graph the second inequality. This is a line with a slope of -1/4 and a y-intercept of 1. Because the inequality is 'greater than or equal to', we will graph a solid line that slopes downward.

Step 5 ：Find the overlapping region. The solution to the system of inequalities is the region where the solutions to both inequalities overlap. This is the region that is above both lines on the graph.

Step 6 ：So, the solution to the system of inequalities is \(y \geq -x/5 - 1\) and \(y \geq -x/4 + 1\).