Question

If the slope of the line joining the points A (x,2) and B (6,-8) is $\dfrac{-5}{4}$, find the value of x.

Answer 1

Hint: Use the fact that the slope of the line joining the points $A\left( {{x}_{1}},{{y}_{1}} \right)$ and $B\left( {{x}_{2}},{{y}_{2}} \right)$ is given by $m=\dfrac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}$ . Substitute the value of ${{x}_{1}},{{x}_{2}},{{y}_{1}},{{y}_{2}}$ in each case and hence find the slope of the line in terms of x. Compare the expression with $\dfrac{-5}{4}$ and hence form an equation in x. Solve for x. Hence find the value of x
Alternatively, assume that the equation of the line is $y=\dfrac{-5}{4}x+c$. Since the line passes through the points, the points satisfy the equation of the line. Substitute (6,-8) in the equation of the line and hence find the value of c. Substitute (x,2) in the equation of line and hence find the value of x.

Complete step-by-step answer:
[i] We have $A\equiv \left( x,2 \right)$ and $B\equiv \left( 6,-8 \right)$
We know that the slope of the line joining the points $A\left( {{x}_{1}},{{y}_{1}} \right)$ and $B\left( {{x}_{2}},{{y}_{2}} \right)$ is given by $m=\dfrac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}$.
Here ${{x}_{1}}=x,{{x}_{2}}=6,{{y}_{1}}=2$ and ${{y}_{2}}=-8$
Hence, we have
$m=\dfrac{-8-2}{6-x}=\dfrac{-10}{6-x}$
But the slope of the line is $\dfrac{-5}{4}$
Hence, we have
$\dfrac{-10}{6-x}=\dfrac{-5}{4}$
Cross multiplying, we get
$-40=-30+5x$
Adding 30 on both sides of the equation, we get
$5x=-10$
Dividing by 5 on both sides of the equation, we get
$x=-2$
Hence the value of x is -2.
Note: Alternative solution:
Let the equation of the line be $y=\dfrac{-5x}{4}+c$
Since (6,-8) lies on the line, we have
$-8=\dfrac{-5}{4}\left( 6 \right)+c$
Multiplying both sides by 4, we get
$-32=-30+4c\Rightarrow c=\dfrac{-1}{2}$
Since (x,2) lies on the line, we have
$2=\dfrac{-5x}{4}+\dfrac{-1}{2}$
Adding $\dfrac{1}{2}$ on both sides, we get
$\dfrac{5}{2}=\dfrac{-5x}{4}$
Multiplying both sides of the equation by $\dfrac{-4}{5}$, we get
$x=\dfrac{5}{2}\times \dfrac{-4}{5}=-2$
Hence the value of x is -2.