Given Information:
- Two lines: 2x - y = 5 and x + 3y - 8 = 0
- Parallel line: 3x - 4y = 7

Step 1: Find the Point of Intersection
To find the point of intersection between the two lines, we can solve the given system of equations:
2x - y = 5
x + 3y - 8 = 0

Solving these equations simultaneously, we get:
2x - y = 5
x + 3y - 8 = 0

Rearranging the second equation, we get:
x = 8 - 3y

Substituting the value of x in the first equation, we have:
2(8 - 3y) - y = 5
16 - 6y - y = 5
-7y = -11
y = -11/-7
y = 11/7

Substituting the value of y in the second equation, we get:
x + 3(11/7) - 8 = 0
x + 33/7 - 8 = 0
x + 33/7 - 56/7 = 0
x - 23/7 = 0
x = 23/7

Therefore, the point of intersection is (23/7, 11/7).

Step 2: Determine the Slope of the Parallel Line
The given parallel line has the equation 3x - 4y = 7. To determine its slope, we can rearrange the equation in slope-intercept form (y = mx + b):
-4y = -3x + 7
y = (3/4)x - 7/4

The slope of the parallel line is 3/4.

Step 3: Find the Equation of the Parallel Line
Since the parallel line has the same slope as the given line, we can use the point-slope form of a line to find its equation. The point-slope form is given by: y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope.

Using the point of intersection (23/7, 11/7) and the slope 3/4, we have:
y - (11/7) = (3/4)(x - 23/7)

Expanding and rearranging the equation, we get:
4y - 44/7 = 3x - 69/7
4y = 3x - 69/7 + 44/7
4y = 3x - 25/7

Therefore, the equation of the line passing through the point of intersection of 2x - y = 5 and x + 3y - 8 = 0 and parallel to the line 3x - 4y = 7 is 4y = 3x - 25/7.