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We are tasked with finding the area enclosed by the x-axis and the straight lines:

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

Let's break down the process step by step:

### 1. Convert the equations into slope-intercept form:
- For \( x = 3y - 5 \), solve for \( y \):
\[
y = \frac{x + 5}{3}
\]
This is the equation of the first line in the form \( y = f(x) \).

- For \( 2y = 7 + 3x \), solve for \( y \):
\[
y = \frac{7 + 3x}{2}
\]
This is the equation of the second line in the form \( y = g(x) \).

### 2. Find the points of intersection:
The area is enclosed between these two lines and the x-axis. To find the limits of integration, we need the points where the two lines intersect and where each line intersects the x-axis (i.e., where \( y = 0 \)).

- First, set \( y = 0 \) in \( x = 3y - 5 \):
\[
x = -5
\]
Thus, the first line intersects the x-axis at \( x = -5 \).

- Now, for \( 2y = 7 + 3x \), set \( y = 0 \):
\[
0 = 7 + 3x \quad \Rightarrow \quad x = -\frac{7}{3} \approx -2.33
\]
Thus, the second line intersects the x-axis at \( x = -\frac{7}{3} \).

Next, find the point where the two lines intersect by setting the equations for \( y \) equal to each other:
\[
\frac{x + 5}{3} = \frac{7 + 3x}{2}
\]
Solve for \( x \). Multiply both sides by 6 to eliminate the denominators:
\[
2(x + 5) = 3(7 + 3x)
\]
Simplify:
\[
2x + 10 = 21 + 9x
\]
\[
10 - 21 = 9x - 2x
\]
\[
-11 = 7x \quad \Rightarrow \quad x = -\frac{11}{7} \approx -1.57
\]

Now substitute \( x = -\frac{11}{7} \) into one of the original equations (say, \( y = \frac{x + 5}{3} \)) to find the corresponding \( y \):
\[
y = \frac{-\frac{11}{7} + 5}{3} = \frac{\frac{35 - 11}{7}}{3} = \frac{24}{7} \times \frac{1}{3} = \frac{8}{7}
\]
Thus, the point of intersection is \( \left( -\frac{11}{7}, \frac{8}{7} \right) \).

### 3. Set up the integral:
The area between the two curves is found by integrating the difference between the two functions from \( x = -5 \) to \( x = -\frac{7}{3} \). The integral is:
\[
\text{Area} = \int_{-5}^{-\frac{7}{3}} \left( \frac{7 + 3x}{2} - \frac{x + 5}{3} \right) dx
\]

Let's calculate this next.

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 x = 3y - 5 and 2y = 7 + 3x through algebraic methods and integration. This problem includes solving linear equations, finding intersections, and integrating between the limits of the curves.

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