Question

Find the slope of the line which passes through the points
[i] (2,5) and (-4,-4) [ii] (-2,3) and (4,-6)

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 slopes of the lines.
Alternatively, assume that the equation of the line is $y=mx+c$. Since the line passes through the points, the points satisfy the equation of the line. Hence form two linear equations in two variables m and c. Solve for m and c. The value of m gives the slope of the line.

Complete step-by-step answer:
[i] We have $A\equiv \left( 2,5 \right)$ and $B\equiv \left( -4,-4 \right)$
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 ${{x}_{1}}=2,{{x}_{2}}=-4,{{y}_{1}}=5$ and ${{y}_{2}}=-4$
Hence, we have
$m=\dfrac{-4-5}{-4-2}=\dfrac{-9}{-6}=\dfrac{3}{2}$
Hence the slope of the line is $\dfrac{3}{2}$
[ii] We have $A\equiv \left( -2,3 \right)$ and $B\equiv \left( 4,-6 \right)$
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 ${{x}_{1}}=-2,{{x}_{2}}=4,{{y}_{1}}=3$ and ${{y}_{2}}=-6$
Hence, we have
$m=\dfrac{-6-3}{4-\left( -2 \right)}=\dfrac{-9}{6}=\dfrac{-3}{2}$
Hence the slope of the line is $\dfrac{-3}{2}$.
Note: Alternative solution:
[i] Let the equation of the line passing through the given points be y = mx+c
Since (2,5) lies on the line, we have
$2m+c=5$
Also since (-4,-4) lies on the line, we have
$-4m+c=-4$
Hence, we have $2m+4m=5+4\Rightarrow m=\dfrac{9}{6}=\dfrac{3}{2}$
[ii] Let the equation of the line passing through the given points be y = mx+c
Since (-2,3) lies on the line, we have
$-2m+c=3$
Also since (4,-6) lies on the line, we have
$4m+c=-6$
Hence, we have $4m+2m=-6-3\Rightarrow m=\dfrac{-3}{2}$.