Question

which graph shows the solution to the system below?
\

$$\begin{cases} x - y > 5 \\\\ 3x + 4y \\leq 12 \\end{cases}$$

a.
graph a
b.
graph b
c.
graph c
d.
graph d

Explanation:

Step1: Analyze \( x - y > 5 \)

Rewrite as \( y < x - 5 \). The boundary line \( y = x - 5 \) has a slope of 1 and y-intercept -5. Since it's \( y < x - 5 \), the line is dashed, and we shade below it.

Step2: Analyze \( 2x + 4y \leq 12 \)

Simplify: \( y \leq -\frac{1}{2}x + 3 \). The boundary line \( y = -\frac{1}{2}x + 3 \) has slope -1/2 and y-intercept 3. It's a solid line, and we shade below it.

Step3: Find the overlapping region

We need the area shaded by both inequalities. Check the options:

• Option A: Shading doesn't match both inequalities.
• Option B: Shading doesn't match \( y < x - 5 \) (shaded above in parts).
• Option C: Shading for \( y < x - 5 \) (below dashed line) and \( y \leq -\frac{1}{2}x + 3 \) (below solid line) overlaps correctly.
• Option D: Shading for \( y < x - 5 \) is incorrect (shaded above).

Answer:

C