https://file.aiquickdraw.com/mathbot/file/1726273831608ccww4r8r.jpg

We are tasked with finding the area enclosed by the **x-axis** and the lines given by the equations:

1. \( x = 3y - 5 \)
2. \( 2y = 7 + 3x \)

Let's begin by analyzing the system of equations and find the points of intersection, which will help us determine the bounds of the area.

### Step 1: Convert the equations to \( y = f(x) \) form.

For \( x = 3y - 5 \):

\[
x = 3y - 5 \implies y = \frac{x + 5}{3}
\]

For \( 2y = 7 + 3x \):

\[
2y = 7 + 3x \implies y = \frac{7 + 3x}{2}
\]

So, the equations in terms of \( y \) are:
1. \( y = \frac{x + 5}{3} \)
2. \( y = \frac{7 + 3x}{2} \)

### Step 2: Find the intersection points.

Now we solve for the points where these lines intersect the **x-axis** by setting \( y = 0 \).

For the first equation \( y = \frac{x + 5}{3} \):

\[
0 = \frac{x + 5}{3} \implies x = -5
\]

For the second equation \( y = \frac{7 + 3x}{2} \):

\[
0 = \frac{7 + 3x}{2} \implies 3x = -7 \implies x = -\frac{7}{3}
\]

### Step 3: Determine the enclosed area.

To find the area between these curves and the **x-axis**, we need to integrate the difference between the upper and lower curves across the interval from \( x = -5 \) to \( x = -\frac{7}{3} \).

The area \( A \) is given by:

\[
A = \int_{-5}^{-\frac{7}{3}} \left[ \frac{7 + 3x}{2} - \frac{x + 5}{3} \right] dx
\]

I'll now calculate this integral to find the area.The area enclosed by the x-axis and the two lines \( x = 3y - 5 \) and \( 2y = 7 + 3x \) is approximately **6.52 square units**.

Would you like further details on how this integral was computed or have any other questions?

Here are 5 related questions you might find useful:

1. How do we determine the limits of integration in such problems?
2. Can we find the area between curves when the x-axis is not involved?
3. How do we solve a system of linear equations geometrically?
4. What happens if the curves intersect at more than two points?
5. How would the area change if the curves were shifted vertically?

**Tip:** To verify your results graphically, plotting the equations can help you visualize the enclosed area more clearly.

Find the area enclosed by the x-axis and the straight lines x = 3y - 5 and 2y = 7 + 3x.

Learn how to calculate the area enclosed by the x-axis and the lines given by x = 3y - 5 and 2y = 7 + 3x using integration. This problem involves coordinate geometry, converting equations, and applying definite integrals.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation