Introduction:
We are given two lines: x + y = 4 and 4x + 3y = 10. The task is to find the number of points on the line x + y = 4 that are unit distance apart from the line 4x + 3y = 10.

Understanding the problem:
To solve this problem, we need to find the points on the line x + y = 4 that are exactly one unit away from the line 4x + 3y = 10. In other words, we need to find the intersection points of these two lines where the distance between them is equal to one unit.

Approach:
To find the intersection points, we can solve the given system of equations simultaneously. Let's solve them step by step:

Step 1: Convert the equations to slope-intercept form.
- Equation 1: x + y = 4 can be rewritten as y = -x + 4
- Equation 2: 4x + 3y = 10 can be rewritten as y = (-4/3)x + 10/3

Step 2: Find the slopes of the lines.
- The slope of Equation 1 is -1.
- The slope of Equation 2 is -4/3.

Step 3: Find the y-intercepts of the lines.
- The y-intercept of Equation 1 is 4.
- The y-intercept of Equation 2 is 10/3.

Step 4: Find the coordinates of the intersection point.
- To find the intersection point, we equate the two equations and solve for x and y.
- Equating y from both equations, we get:
-x + 4 = (-4/3)x + 10/3
-3x + 12 = -4x + 10
x = 2
- Substituting the value of x in either equation, we get:
y = -2 + 4
y = 2
- Therefore, the intersection point is (2, 2).

Step 5: Calculate the distance between the intersection point and the line.
- We can use the distance formula to calculate the distance between a point (x1, y1) and a line Ax + By + C = 0.
- The distance formula is given by:
d = |Ax1 + By1 + C| / sqrt(A^2 + B^2)
- Substituting the values, we have:
d = |4(2) + 3(2) - 10| / sqrt(4^2 + 3^2)
d = |-16| / sqrt(16 + 9)
d = 16 / sqrt(25)
d = 16 / 5
d = 3.2 units

Conclusion:
There is only one point on the line x + y = 4 that is exactly one unit away from the line 4x + 3y = 10. The coordinates of this point are (2, 2), and the distance between the point and the line is 3.2 units.

There must be 2 points