Find a if the point (3, a) is on the line that passes through (-2, 7) and (5, -3).

Question:

Find {eq}a {/eq} if the point {eq}(3, a) {/eq} is on the line that passes through {eq}(-2, 7) {/eq} and {eq}(5, -3) {/eq}.

Slope of a Line:

The slope of a line is by how much the line is rising according to the run. The slope of a line that passes through the points {eq}(x_1, y_1) {/eq} and {eq}(x_2, y_2) {/eq} is given by:

$$m= \dfrac{y_2-y_1}{x_2-x_1} $$

Answer and Explanation:

The slope of a line through two points {eq}(x_1, y_1) {/eq} and {eq}(x_2,y_2) {/eq} is calculated using:

$$m= \dfrac{y_2-y_1}{x_2-x_1} $$

The given 3 points on the line are:

$$A(3, a); \,\, B (-2, 7);\,\, C(5, -3) $$

Since these 3 points lie on the same line:

$$\begin{align} \text{Slope of AB}&= \text{Slope of BC} \\ \dfrac{7-a}{-2-3}&= \dfrac{-3-7}{5-(-2)} & \text{(Using slope formula)}\\ \dfrac{7-a}{-5} &= \dfrac{-10}{7} \\ 7(7-a) &= (-5)(-10) & \text{(Cross multipled)} \\ 49-7a& =50 \\ -7a &= 1 & \text{(Subtracted 49 from both sides)} \\ a&= \boxed{\mathbf{ \dfrac{-1}{7}}} & \text{(Divided both sides by -7)} \end{align} $$