Let's analyze the problem by finding the equations of the lines and solving the region enclosed by them:

Step 1: Equations of the Lines

The equations given are:

1. $2x - y = 3$, which we can rewrite as: $y = 2x - 3$
2. $y = -x$
3. $y = 7$

Step 2: Find the Points of Intersection

We need to find the points of intersection of the three lines to define the vertices of the region enclosed.

• Intersection of $y = 2x - 3$ and $y = -x$: $2x - 3 = -x$ Solving for $x$: $3x = 3 \implies x = 1$ Substituting $x = 1$ into $y = -x$: $y = -1$ So, the intersection point is $(1, -1)$.

• Intersection of $y = 2x - 3$ and $y = 7$: $7 = 2x - 3$ Solving for $x$: $2x = 10 \implies x = 5$ So, the intersection point is $(5, 7)$.

• Intersection of $y = -x$ and $y = 7$: $7 = -x$ Solving for $x$: $x = -7$ So, the intersection point is $(-7, 7)$.

Step 3: Vertices of the Triangle

The vertices of the triangle formed by the three lines are:

1. $(1, -1)$
2. $(5, 7)$
3. $(-7, 7)$

Step 4: Calculate the Area

We can use the formula for the area of a triangle given its vertices $(x_1, y_1), (x_2, y_2), (x_3, y_3)$: $\text{Area} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right|$

Substitute the coordinates $(1, -1), (5, 7), (-7, 7)$: $\text{Area} = \frac{1}{2} \left| 1(7 - 7) + 5(7 - (-1)) + (-7)(-1 - 7) \right|$ $= \frac{1}{2} \left| 1(0) + 5(8) + (-7)(-8) \right|$ $= \frac{1}{2} \left| 0 + 40 + 56 \right| = \frac{1}{2} \times 96 = 48$

So, the area of the enclosed region is 48 square units.

Would you like further details or explanations on any step?

Here are 5 related questions to consider:

1. How do you find the intersection points of two lines algebraically?
2. What other methods can you use to find the area of a triangle given its vertices?
3. How would the process change if one of the equations were a curve, like $y = x^2$?
4. Can you use a geometric approach to estimate the area of this triangle without algebra?
5. How would graphing the equations help visualize the solution?

Tip: Always double-check the intersection points, as they form the basis for calculating the enclosed region's area.