The problem asks us to graph the solution set of the following system of inequalities and find the vertices of the solution region: $y \le x$ $x + y \ge 3$ $x \le 7$

1. Problem Description

The problem asks us to graph the solution set of the following system of inequalities and find the vertices of the solution region:

$y \le x$

$x + y \ge 3$

$x \le 7$

2. Solution Steps

First, we rewrite the inequalities as equations to find the boundary lines:

$y = x$

$x + y = 3$

$x = 7$

Next, we find the intersection points of these lines.

Intersection of $y = x$ and $x + y = 3$:

Substitute $y = x$ into the second equation:

$x + x = 3$

$2x = 3$

$x = \frac{3}{2}$

Since $y = x$, $y = \frac{3}{2}$

So the intersection point is $(\frac{3}{2}, \frac{3}{2})$

Intersection of $y = x$ and $x = 7$:

Since $x = 7$, $y = 7$

So the intersection point is $(7, 7)$

Intersection of $x + y = 3$ and $x = 7$:

Substitute $x = 7$ into the second equation:

$7 + y = 3$

$y = 3 - 7$

$y = -4$

So the intersection point is $(7, -4)$

Now we consider the inequalities.

$y \le x$: This represents the region below the line $y = x$.

$x + y \ge 3$: This represents the region above the line $x + y = 3$.

$x \le 7$: This represents the region to the left of the line $x = 7$.

The vertices of the solution region are the intersection points of the boundary lines that satisfy all three inequalities. We found three intersection points: $(\frac{3}{2}, \frac{3}{2})$, $(7, 7)$, and $(7, -4)$. Since $x \le 7$, all points to the right of $x=7$ are excluded.

$(\frac{3}{2}, \frac{3}{2})$:

$y \le x$: $\frac{3}{2} \le \frac{3}{2}$ (True)

$x + y \ge 3$: $\frac{3}{2} + \frac{3}{2} \ge 3$, $3 \ge 3$ (True)

$x \le 7$: $\frac{3}{2} \le 7$ (True)

So, $(\frac{3}{2}, \frac{3}{2})$ is a vertex.

$(7, 7)$:

$y \le x$: $7 \le 7$ (True)

$x + y \ge 3$: $7 + 7 \ge 3$, $14 \ge 3$ (True)

$x \le 7$: $7 \le 7$ (True)

So, $(7, 7)$ is a vertex.

$(7, -4)$:

$y \le x$: $-4 \le 7$ (True)

$x + y \ge 3$: $7 + (-4) \ge 3$, $3 \ge 3$ (True)

$x \le 7$: $7 \le 7$ (True)

So, $(7, -4)$ is a vertex.

3. Final Answer

The vertices of the solution are $(\frac{3}{2}, \frac{3}{2})$, $(7, 7)$, and $(7, -4)$.