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

To find the area enclosed by the \( x \)-axis and the two given lines, we can follow these steps:

### Step 1: Rewrite the equations in slope-intercept form
1. For the line \( x = 3y - 5 \):
   \[
   y = \frac{x + 5}{3}
   \]

2. For the line \( 2y = 7 + 3x \):
   \[
   y = \frac{3}{2}x + \frac{7}{2}
   \]

### Step 2: Find the points of intersection
We'll find the intersection points of these lines with each other and the \( x \)-axis.

1. **Intersection of the two lines**:
   Set \( \frac{x + 5}{3} = \frac{3}{2}x + \frac{7}{2} \):
   \[
   2(x + 5) = 3(3x + 7)
   \]
   Expanding and solving, we find \( x = -\frac{11}{7} \).

   Substituting \( x = -\frac{11}{7} \) back into either equation for \( y \) gives \( y = -\frac{8}{7} \).

   So, the intersection point of the lines is \( \left(-\frac{11}{7}, -\frac{8}{7}\right) \).

2. **Intersection of each line with the \( x \)-axis**:
   - For \( x = 3y - 5 \): Set \( y = 0 \) and solve for \( x \):
     \[
     x = -5
     \]
     So, this line intersects the \( x \)-axis at \( (-5, 0) \).

   - For \( 2y = 7 + 3x \): Set \( y = 0 \) and solve for \( x \):
     \[
     x = -\frac{7}{3}
     \]
     So, this line intersects the \( x \)-axis at \( \left(-\frac{7}{3}, 0\right) \).

### Step 3: Set up the integral to find the area
The area between the curves from \( x = -5 \) to \( x = -\frac{7}{3} \) is:
\[
\text{Area} = \int_{-5}^{-\frac{7}{3}} \left(\frac{x + 5}{3} - \left(\frac{3}{2}x + \frac{7}{2}\right)\right) \, dx
\]

### Step 4: Evaluate the integral
1. Simplify the integrand:
   \[
   \frac{x + 5}{3} - \left(\frac{3}{2}x + \frac{7}{2}\right) = -\frac{11}{6}x - \frac{17}{6}
   \]

2. Integrate term-by-term:
   \[
   \text{Area} = \int_{-5}^{-\frac{7}{3}} \left(-\frac{11}{6}x - \frac{17}{6}\right) \, dx
   \]

Evaluating this integral will yield the area.

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

Learn how to find the area enclosed by the x-axis and the lines x=3y-5 and 2y=7+3x. This solution covers key geometry and calculus concepts, including slope-intercept forms and integration. Ideal for students in Grades 11-12.

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