William H. answered • 04/18/20

Patient and Effective SAT Math and English Tutor

It is easiest to begin by graphing the line segment AB = (-4,8),(6,3).

The ratio of **3/2** requires dividing the line segment AB into fifths, with three fifths before (to the left of) the point P and two fifths after point P.

The trick here is to realize that the line can be divided properly by looking at either one of the coordinate axes.

Using the x-axis, we see that the line segment covers 10 units (**-4** to **6**). So, we need to position P so that there are six units before it and four units after it (6/4 = **3/2**). So, the x-value of P is 2 (2-(**-4**) = 6; **6**-2 = 4; 6/4 = **3/2**).

The y-coordinate of P can be derived similarly by using the y-axis. However, the answer requirements refer to m, which, in linear equations, is used to refer to the slope of the line. So, let's get the slope-intercept form of the equation for AB and use that to find the y-value of P.

The slope of AB is (3-8)/(6-(-4)) = -5/10 = -1/2. The y-intercept is 6. We know that because the y-intercept occurs when x=0, which is 4/10 (or 2/5) of the distance from A to B along the x-axis, so the intercept is 2/5 of the distance from A to B along the y-axis, that is, running downward from y=8 to y=3; y=6 is 2/5 the distance from y=8 to y=3.

So, we have the slope-intercept form of the line segment AB: y = (-1/2)x + 6, with -4 ≤ x ≤ 6.

We plug in x = 2 to get y = 5; hence, P = (2,5).

Instead of deriving the x-coordinate of P using the x-axis, we could have derived the y-coordinate P using the y-axis. The segment AB goes from y = 8 to y = 3, which is five units, and 3/5 of five units is three units, so the y-value of P is three units down from 8, at y = 5. You can then derive the x-value of P using the same slope-intercept formula above.