Question

which graph shows the solution to the system of inequalities below? \\(\

$$\begin{cases} x + y < 3 \\\\ -5x + 2y \\leq 10 \\end{cases}$$

\\)

Explanation:

Step1: Analyze \( x + y < 3 \)

Rewrite as \( y < -x + 3 \). The line \( y = -x + 3 \) has a slope of \(-1\) and y - intercept \( 3 \). Since the inequality is \( < \), the line is dashed, and we shade below it.

Step2: Analyze \( -5x + 2y \leq 10 \)

Rewrite as \( 2y \leq 5x + 10 \) or \( y \leq \frac{5}{2}x + 5 \). The line \( y=\frac{5}{2}x + 5 \) has a slope of \( \frac{5}{2} \) and y - intercept \( 5 \). Since the inequality is \( \leq \), the line is solid, and we shade below it.

Step3: Check the graph

The given graph has a solid line (for \( -5x + 2y \leq 10 \)) with y - intercept \( 5 \) and a dashed line (for \( x + y < 3 \)) with y - intercept \( 3 \). The shaded region is the intersection of the regions below \( y < -x + 3 \) and below \( y \leq \frac{5}{2}x + 5 \), which matches the solution of the system.

Answer:

The graph shown (the one with the dashed line \( x + y = 3 \), solid line \( - 5x+2y = 10 \), and the shaded region as in the provided image) is the solution graph.