P is a point on the graph of y 5x 3  . The coordinates of a point Q are (3, –2). If M is the midpoint of PQ, then M must lie on the line represented by

To find the line on which the midpoint M of PQ lies, we need to first find the coordinates of point P.

Given that P is a point on the graph of y = 5x + 3, we can substitute the x-coordinate of Q (which is 3) into the equation to find the y-coordinate of P.

Substituting x = 3 into the equation y = 5x + 3, we get: y = 5(3) + 3 y = 15 + 3 y = 18

So, the coordinates of point P are (3, 18).

Now, we can find the coordinates of the midpoint M by using the midpoint formula. The midpoint formula states that the x-coordinate of the midpoint is the average of the x-coordinates of the two points, and the y-coordinate of the midpoint is the average of the y-coordinates of the two points.

Using the coordinates of points P and Q, we can calculate the coordinates of M as follows:

x-coordinate of M = (x-coordinate of P + x-coordinate of Q) / 2 = (3 + 3) / 2 = 6 / 2 = 3

y-coordinate of M = (y-coordinate of P + y-coordinate of Q) / 2 = (18 + (-2)) / 2 = 16 / 2 = 8

Therefore, the coordinates of the midpoint M are (3, 8).

To determine the line on which M lies, we can use the slope-intercept form of a line, y = mx + b, where m is the slope and b is the y-intercept.

Since M lies on the line passing through points P and Q, we can calculate the slope of the line using the formula:

m = (y-coordinate of Q - y-coordinate of P) / (x-coordinate of Q - x-coordinate of P) = (-2 - 18) / (3 - 3) = -20 / 0

The slope of the line is undefined because the denominator is zero.

When the slope of a line is undefined, it means that the line is vertical. Therefore, the line on which the midpoint M of PQ lies is a vertical line passing through the point (3, 8).