To find the equation of the straight line passing through the point (4, 3) and making equal intercepts on the coordinate axes, we can use the slope-intercept form of a line, which is given by the equation y = mx + b, where m is the slope of the line and b is the y-intercept.

Finding the slope:
Since the line makes equal intercepts on the coordinate axes, we know that the x-intercept and y-intercept are equal in magnitude. Let's call this intercept value 'a'.

The x-intercept is the point where the line intersects the x-axis, which can be written as (a, 0).
The y-intercept is the point where the line intersects the y-axis, which can be written as (0, a).

Using the two points (4, 3) and (a, 0), we can find the slope of the line using the formula:

m = (y2 - y1) / (x2 - x1)
m = (0 - 3) / (a - 4)
m = -3 / (a - 4)

Finding the y-intercept:
Since the line passes through the point (4, 3), we can substitute these values into the slope-intercept equation and solve for the y-intercept:

3 = (-3 / (a - 4)) * 4 + b
3 = -12 / (a - 4) + b
3(a - 4) = -12 + b(a - 4)
3a - 12 = -12 + b(a - 4)
3a - 12 = -12 + ab - 4b
3a - ab + 4b = 0

Since we know that the x-intercept and y-intercept are equal, we can set a = b:

3a - a^2 + 4a = 0
3a + 4a - a^2 = 0
7a - a^2 = 0
a(7 - a) = 0

Therefore, a = 0 or a = 7.

If a = 0, it means that the line passes through the origin, which does not satisfy the condition of making equal intercepts on the coordinate axes.

If a = 7, it means that the line intersects the x-axis at (7, 0) and the y-axis at (0, 7), which satisfies the condition of making equal intercepts on the coordinate axes.

Thus, the equation of the line passing through the point (4, 3) and making equal intercepts on the coordinate axes is:
x + y = 7

Therefore, the correct answer is option A: x + y = 7.