Question

If the slope of the line joining $$\mathrm{P}(k, 3)$$ and $$\mathrm{Q}(8, -6)$$ is $$\frac{-3}{4}$$, find the value of $$k$$.

Answer 1

The formula for the slope of a line joining two points is $$m = \frac{y_2 - y_1}{x_2 - x_1}$$

Here, $$(x_1, y_1) = (k, 3)$$, $$(x_2, y_2) = (8, -6)$$ and $$m = -\frac{3}{4}$$.

Substitute the values:

$$-\frac{3}{4} = \frac{-6 - 3}{8 - k}$$

$$-\frac{3}{4} = \frac{-9}{8 - k}$$

$$\frac{3}{4} = \frac{9}{8 - k}$$

Cross-multiplying gives:

$$3(8 - k) = 4 \times 9$$ 

$$24 - 3k = 36$$

$$-3k = 36 - 24$$

$$-3k = 12$$ 

$$k = \frac{12}{-3} = -4$$

Thus, the value of $$k$$ is $$-4$$.