In a rectangular coordinate system, which of the following points is intersected by the line connected by the coordinates (5, 6) and (21, 18)?

A. (9, 9)
B. (12, 12)
C. (13, 13)
D. (12, 13)
E. (16, 15)

(A) To solve this problem, we first need to determine the slope of the line by substituting values into the slope equation.

Slope = change in y / change in x = (y2 – y1)/(x2 – x1)

Therefore, slope = (18 – 6)/(21 – 5) = 12/16 = 3/4 .

Now that we have the slope, we can determine the equation of the line. The equation for a line is y = mx + b, where m is the slope and b is the y-intercept.

Since m = 3/4, we can solve for b by plugging in the point (5, 6) for the values of x and y:
y = mx + b
6 = (3/4)(5) + b
6 = 15/4 + b
6 – 15/4 = b
9/4 = b

So the equation for the line is:
y = (3/4)x + 9/4

Only the (x, y) pair that makes the equation true will be a point on the line described by the equation above. When we substitute the values of x and y from the coordinates in the answer choices, only choice (A) makes the equation true:
9 = (3/4)(9) + 9/4
9 = 27/4 + 9/4 = 36/4 = 9.

The correct answer is choice (A). This question is also easy to solve by simply back-solving the answer choices.
----------

12 and 13 also works:
12 = (3/4) × 13 + 9/4
12 = 39/4 + 9/4
12 = 48/4
12 = 12

Here y = 12 and x = 13. You have used point (13, 12), NOT (12, 13), which would be
13 ?= (3/4) × 12 + 9/4
This does not fit.