https://file.aiquickdraw.com/mathbot/file/17265072531476b4osqij.jpg

The problem asks to find the area of the region enclosed by the line \(4x + 3y = 12\), the x-axis, and the y-axis.

### Steps to solve:

1. **Find the intercepts of the line**:
   - The x-intercept occurs when \(y = 0\). Substituting \(y = 0\) into the equation \(4x + 3(0) = 12\), we get:
     \[
     4x = 12 \quad \Rightarrow \quad x = 3
     \]
     So, the x-intercept is at \((3, 0)\).
   - The y-intercept occurs when \(x = 0\). Substituting \(x = 0\) into the equation \(4(0) + 3y = 12\), we get:
     \[
     3y = 12 \quad \Rightarrow \quad y = 4
     \]
     So, the y-intercept is at \((0, 4)\).

2. **Plot the line and identify the enclosed region**:
   The region is a right triangle with vertices at \((0, 0)\), \((3, 0)\), and \((0, 4)\), where the x-intercept and y-intercept form the base and height of the triangle, respectively.

3. **Calculate the area of the triangle**:
   The area of a triangle is given by the formula:
   \[
   \text{Area} = \frac{1}{2} \times \text{base} \times \text{height}
   \]
   Here, the base is 3 (the x-intercept) and the height is 4 (the y-intercept), so:
   \[
   \text{Area} = \frac{1}{2} \times 3 \times 4 = 6 \, \text{square units}
   \]

Thus, the area of the region enclosed by the line, the x-axis, and the y-axis is **6 square units**.

Let me know if you'd like further clarification or details!

Here are 5 questions that expand on this concept:
1. How would the area change if the equation of the line was \(2x + 3y = 6\)?
2. Can you calculate the slope of the line \(4x + 3y = 12\)?
3. What would be the area of the enclosed region if the line passed through \((0, 6)\) instead of \((0, 4)\)?
4. How can the distance between the x-intercept and y-intercept be calculated?
5. What other methods can be used to find the area of a triangle besides the base-height formula?

**Tip**: Always remember that the x- and y-intercepts are key in identifying the area of triangles formed with linear equations.

By plotting the graph of 4x + 3y = 12 on the grid below, work out the area of the region enclosed between the line 4x + 3y = 12, the x-axis, and the y-axis. Give your answer in square units.

Learn how to calculate the area enclosed by the line 4x + 3y = 12, the x-axis, and the y-axis using basic geometry and linear equations. This problem is suitable for students in Grades 8-10 and involves calculating the intercepts and finding the area of a triangle.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation