Final answer:
The slope of the line perpendicular to the given line x + 2y = 5 is 2. The equation of this line passing through the point (4,10) is y = 2x + 2.
Step-by-step explanation:
To find the slope of a line that is perpendicular to a given line, first, we need to find the slope of the given line. For the line represented by the equation x + 2y = 5, we can rearrange it into y = mx + b form to determine the slope, m, where m is the slope of the line and b is the y-intercept. The original line equation can be transformed to 2y = -x + 5, or y = (-x/2) + 5/2, so the slope of this line is -1/2. The slope of a line perpendicular to this line will be the negative reciprocal, so the new slope will be 2. Consequently, the slope of the line passing through the point (4,10) and perpendicular to the x + 2y = 5 line is 2.
Now, using the point-slope formula, which is y - y1 = m(x - x1), where (x1, y1) is the point the line passes through and m is the slope, we plug in (4,10) for (x1, y1) and 2 for m. The equation becomes y - 10 = 2(x - 4). Simplifying this, we get:
y - 10 = 2x - 8
y = 2x + 2
This is the equation of the line that is perpendicular to the original line and passes through the point (4,10).