To find the equation of the bisector of the angle between the lines x + 2y = 3 and 3x - 4y = 1, we can follow these steps:

1. Find the slopes of the given lines.
The slope of the first line (x + 2y = 3) can be found by rearranging the equation into slope-intercept form (y = mx + b):
x + 2y = 3
2y = -x + 3
y = -0.5x + 1.5
Comparing with the equation y = mx + b, we can see that the slope (m1) of this line is -0.5.

The slope of the second line (3x - 4y = 1) can also be found by rearranging the equation into slope-intercept form:
3x - 4y = 1
-4y = -3x + 1
y = 0.75x - 0.25
Comparing with the equation y = mx + b, we can see that the slope (m2) of this line is 0.75.

2. Calculate the average of the slopes.
The average of the slopes is given by (m1 + m2) / 2:
(-0.5 + 0.75) / 2 = 0.25 / 2 = 0.125

3. Convert the average slope into the tangent of the angle.
The tangent of the angle is given by tan(theta) = m, where m is the slope.
Therefore, tan(theta) = 0.125

4. Find the angle whose tangent is 0.125.
Using an inverse tangent function (tan^(-1)), we can find the angle:
theta = tan^(-1)(0.125)

5. Calculate the slope of the bisector line.
The slope of the bisector line can be found using the tangent sum formula:
tan(2theta) = (2 * tan(theta)) / (1 - tan^2(theta))
Substituting the value of theta:
tan(2 * tan^(-1)(0.125)) = (2 * 0.125) / (1 - (0.125)^2)
tan(2 * tan^(-1)(0.125)) = 0.25 / 0.984375 = 0.253968254
Therefore, the slope of the bisector line is approximately 0.253968254.

6. Find a point on the bisector line.
We can choose any point that lies on both of the given lines. Let's choose the point (1, 0) which lies on both lines.

7. Use the point-slope form to find the equation of the bisector line.
Using the point-slope form (y - y1) = m(x - x1) and substituting the values:
(y - 0) = 0.253968254(x - 1)
y = 0.253968254x - 0.253968254

Therefore, the equation of the bisector of the angle between the lines x + 2y = 3 and 3x - 4y