To find the area of the triangle formed by the intersection of the lines $y = 2x + 5$, $y = x + 5$, and $y = -3$, we first need to find the points where these lines intersect. These points will be the vertices of the triangle.

Step 1: Find points of intersection

1. Intersection of $y = 2x + 5$ and $y = x + 5$: $2x + 5 = x + 5$ $2x - x = 5 - 5$ $x = 0$ Substituting $x = 0$ back into either equation: $y = 2(0) + 5 = 5$ So, the first point of intersection is $(0, 5)$.

2. Intersection of $y = 2x + 5$ and $y = -3$: $2x + 5 = -3$ $2x = -3 - 5$ $2x = -8$ $x = -4$ Substituting $x = -4$ into either equation: $y = -3$ So, the second point of intersection is $(-4, -3)$.

3. Intersection of $y = x + 5$ and $y = -3$: $x + 5 = -3$ $x = -3 - 5 = -8$ Substituting $x = -8$ into either equation: $y = -3$ So, the third point of intersection is $(-8, -3)$.

Step 2: Use the vertices to calculate the area

Now we have the three vertices of the triangle:

• $(0, 5)$
• $(-4, -3)$
• $(-8, -3)$

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

Substituting the coordinates $(0, 5)$, $(-4, -3)$, and $(-8, -3)$: $\text{Area} = \frac{1}{2} \left| 0((-3) - (-3)) + (-4)((-3) - 5) + (-8)(5 - (-3)) \right|$ $= \frac{1}{2} \left| 0(0) + (-4)(-8) + (-8)(8) \right|$ $= \frac{1}{2} \left| 0 + 32 - 64 \right|$ $= \frac{1}{2} \left| -32 \right|$ $= \frac{1}{2} \times 32 = 16$

Final Answer:

The area of the triangle is $\boxed{16}$ square units.

Would you like further details or have any questions?

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Here are 5 related questions to expand on the topic:

1. How do you calculate the area of a triangle if the points are in 3D space?
2. Can you find the centroid of the triangle formed by these three lines?
3. What if the lines formed a quadrilateral instead? How would you find the area?
4. How would you solve this if the equations of the lines were in parametric form?
5. How can we verify that these three lines do indeed form a triangle?

Tip: When dealing with multiple intersecting lines, drawing a sketch can help you visualize the arrangement and prevent mistakes in finding points of intersection.