Solution:

To find the equation of the straight line passing through the point of intersection of two given lines and passing through a given point, we can follow these steps:

1. Find the point of intersection of the given lines.
2. Use the point-slope form of a straight line to find the equation of the line passing through the given point and the point of intersection.

Step 1: Finding the Point of Intersection
We are given two lines:
1) x + 2y - 5 = 0 ....(1)
2) x - 3y - 7 = 0 ....(2)

To find the point of intersection, we can solve these two equations simultaneously.

From equation (1), we can rewrite it as: x = 5 - 2y

Substituting this value of x in equation (2):

5 - 2y - 3y - 7 = 0
-5y - 2 = 0
-5y = 2
y = -2/5

Substituting this value of y in equation (1):

x + 2(-2/5) - 5 = 0
x - 4/5 - 5 = 0
x - 4/5 = 5
x = 5 + 4/5
x = 29/5

So, the point of intersection is (29/5, -2/5).

Step 2: Finding the Equation of the Line
Now, we need to find the equation of the line passing through the point (29/5, -2/5) and the given point (1, 0).

We can use the point-slope form of a straight line, which is given by:
y - y1 = m(x - x1)

where (x1, y1) is a point on the line and m is the slope of the line.

Let's find the slope (m) using the two given points:
m = (y2 - y1) / (x2 - x1)
= (0 - (-2/5)) / (1 - 29/5)
= 2/5 / (-24/5)
= -2/24
= -1/12

Using the point-slope form with the point (29/5, -2/5):

y - (-2/5) = (-1/12)(x - 29/5)
y + 2/5 = (-1/12)(x - 29/5)
y + 2/5 = (-1/12)x + (29/60)
y = (-1/12)x + (29/60) - (12/60)
y = (-1/12)x + (17/60)

Comparing this equation with the given options, we can see that the correct answer is option 'A': x - 12y = 1.