Answer:
For the points P(1, 4), Q(3, y), and R(-3, 16) to be collinear, they must lie on the same straight line. This means that the slopes between any two pairs of these points must be equal.
The formula for calculating the slope (m) between two points (x1, y1) and (x2, y2) is:
\[ m = \frac{{y2 - y1}}{{x2 - x1}} \]
Let's calculate the slopes between these points:
1. Slope between P(1, 4) and Q(3, y):
\[ m_{PQ} = \frac{{y - 4}}{{3 - 1}} = \frac{{y - 4}}{2} \]
2. Slope between Q(3, y) and R(-3, 16):
\[ m_{QR} = \frac{{16 - y}}{{-3 - 3}} = \frac{{16 - y}}{-6} = \frac{{y - 16}}{6} \]
For these points to be collinear, the slopes \(m_{PQ}\) and \(m_{QR}\) must be equal:
\[ \frac{{y - 4}}{2} = \frac{{y - 16}}{6} \]
Now, let's solve for \(y\):
\[ 3(y - 4) = 2(y - 16) \]
\[ 3y - 12 = 2y - 32 \]
\[ y = -32 + 12 \]
\[ y = -20 \]
So, the value of \(y\) for which the points P(1, 4), Q(3, y), and R(-3, 16) are collinear is \(y = -20\).