If points $(t, 2t)$, $(-2, 6)$ and $(3, 1)$ are collinear, then $t$ is equal to

Method 1: Using Slopes (Easiest & Fastest Method) If three points are collinear, then the slope between any two pairs of points must be equal. Let's call our points: - Point A: $(t, 2t)$ - Point B: $(-2, 6)$ - Point C: $(3, 1)$ Find the slope between points B and C: $$\text{Slope of BC} = \frac{1 - 6}{3 - (-2)} = \frac{-5}{5} = -1$$ Find the slope between points A and B: $$\text{Slope of AB} = \frac{6 - 2t}{-2 - t}$$ Set the slopes equal since points are collinear: $$\frac{6 - 2t}{-2 - t} = -1$$ Solve for $t$: Cross multiply: $6 - 2t = -1(-2 - t)$ $$6 - 2t = 2 + t$$ $$6 - 2 = t + 2t$$ $$4 = 3t$$ $$t = \frac{4}{3}$$ Method 2: Using Area Formula (Alternative Method) If three points are collinear, the area of the triangle they form is zero. Using the area formula: $$\text{Area} = \frac{1}{2}|x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)|$$ Where $(x_1, y_1) = (t, 2t)$, $(x_2, y_2) = (-2, 6)$, $(x_3, y_3) = (3, 1)$ Substitute into the formula: $$\frac{1}{2}|t(6 - 1) + (-2)(1 - 2t) + 3(2t - 6)| = 0$$ Simplify inside the absolute value: $$\frac{1}{2}|t(5) + (-2)(1 - 2t) + 3(2t - 6)| = 0$$ $$\frac{1}{2}|5t + (-2 + 4t) + (6t - 18)| = 0$$ $$\frac{1}{2}|5t - 2 + 4t + 6t - 18| = 0$$ $$\frac{1}{2}|15t - 20| = 0$$ Since area equals zero: $$|15t - 20| = 0$$ The absolute value equals zero only when the expression inside equals zero: $$15t - 20 = 0$$ $$15t = 20$$ $$t = \frac{4}{3}$$ Answer: $t = \frac{4}{3}$