Given:
- The equation of line AB is 4x - y = 1.
- Point A is (2,7).
- The equation of line BC is 3x - 4y + 1 = 0.

To Find:
The equation of line AC, such that AB = BC.

Approach:
1. Find the coordinates of point B where line AB intersects line BC.
2. Calculate the distance between points A and B.
3. Calculate the equation of line AC using the midpoint formula and the slope between points A and C.

Solution:

Step 1: Finding Point B
To find the coordinates of point B, we need to solve the system of equations formed by lines AB and BC.

We have the equations:
1. 4x - y = 1
2. 3x - 4y + 1 = 0

Solving these equations simultaneously, we get:
4x - y = 1 (Equation 1)
3x - 4y + 1 = 0 (Equation 2)

Rearranging Equation 2, we have:
3x - 4y = -1

Multiplying Equation 1 by 4 and Equation 2 by -1, we get:
16x - 4y = 4
-3x + 4y = 1

Adding both equations, we eliminate the y variable:
16x - 3x = 4 + 1
13x = 5
x = 5/13

Substituting the value of x into Equation 1, we can find y:
4(5/13) - y = 1
20/13 - y = 1
-y = 1 - 20/13
-y = (13 - 20)/13
-y = -7/13
y = 7/13

Therefore, the coordinates of point B are (5/13, 7/13).

Step 2: Calculating Distance AB
The distance between points A(2,7) and B(5/13, 7/13) can be calculated using the distance formula:

Distance AB = sqrt((x2 - x1)^2 + (y2 - y1)^2)

Plugging in the values, we get:
Distance AB = sqrt((5/13 - 2)^2 + (7/13 - 7)^2)
Distance AB = sqrt((15/13)^2 + (-84/13)^2)
Distance AB = sqrt(225/169 + 7056/169)
Distance AB = sqrt(7281/169)
Distance AB = 85/13

Step 3: Calculating Equation of Line AC
To find the equation of line AC, we need to find the coordinates of point C. Since AB = BC, the distance between points A and C will also be 85/13.

Using the midpoint formula, we can find the coordinates of point C:
Midpoint = ((x1 + x2)/2, (y1 + y2)/2)

Substituting the coordinates of point A and distance AB into the midpoint formula, we get:
((2 + x2)/2, (7 + y2)/2) = (