Solution:

To find the equation of the line through the point of intersection of two given lines and perpendicular to a given line, we need to follow these steps:

1. Find the point of intersection of the given lines.
2. Determine the slope of the given line.
3. Use the negative reciprocal of the slope of the given line to find the slope of the desired line.
4. Use the slope-intercept form of a linear equation (y = mx + b) to find the equation of the desired line.

Step 1: Finding the point of intersection

Given lines:
1. x - 8y + 11 = 0
2. 4x - 7y + 3 = 0

To find the point of intersection, we can solve these two equations simultaneously. By solving these equations, we get x = -1 and y = -1 as the values of the point of intersection.

Therefore, the point of intersection is (-1, -1).

Step 2: Determining the slope of the given line

Given line:
3x - 2y + 5 = 0

To determine the slope of this line, we need to bring it into the slope-intercept form (y = mx + b). By rearranging the equation, we get y = (3/2)x + (5/2).

The slope of the given line is 3/2.

Step 3: Finding the slope of the desired line

The desired line is perpendicular to the given line. Since perpendicular lines have slopes that are negative reciprocals of each other, the slope of the desired line is -2/3.

Step 4: Finding the equation of the desired line

Using the point of intersection (-1, -1) and the slope of the desired line (-2/3), we can plug these values into the slope-intercept form (y = mx + b) to find the equation.

y = (-2/3)x + b

To find the value of b, we substitute the coordinates (-1, -1) into the equation:

-1 = (-2/3)(-1) + b
-1 = 2/3 + b
b = -1 - 2/3
b = -5/3

Therefore, the equation of the line through the point of intersection and perpendicular to the line 3x - 2y + 5 = 0 is:

y = (-2/3)x - 5/3