https://file.aiquickdraw.com/mathbot/file/1726779212622gjrbl3u7.jpg

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.

By plotting the lines 2x - y = 3, y = -x, and y = 7 on a grid, find the area of the region enclosed between the three lines. Give your answer in square units.

Explore how to find the area of a triangular region enclosed by the lines 2x - y = 3, y = -x, and y = 7. This problem involves understanding linear equations, finding intersections, and applying the area formula for a triangle. Suitable for students in Grades 8-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation