Question

How do you find the slope given $\left( 3,2 \right),\ \left( 5,12 \right)$?

Answer 1

From the question given we have been asked to find the slope of the two points they are $\left( 3,2 \right),\ \left( 5,12 \right)$. We know that the formula for the slope of two points if $\left( {{x}_{1}},{{y}_{1}} \right),\ \left( {{x}_{2}},{{y}_{2}} \right)$ then the slope of these two points is $m=\dfrac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}$. By this formula we will get the slope of the two points.

Complete step-by-step solution:
From the given question we have been asked to find the slope of the two points
$\Rightarrow \left( 3,2 \right),\ \left( 5,12 \right)$
As we know that the formula for the slope of two points if $\left( {{x}_{1}},{{y}_{1}} \right)\ \left( {{x}_{2}},{{y}_{2}} \right)$ then the slope of these two points is
$\Rightarrow m=\dfrac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}$
By comparing the given with the general points we will get the values as
Here the values are
$\Rightarrow {{x}_{1}}=3$
$\Rightarrow {{x}_{2}}=5$
$\Rightarrow {{y}_{1}}=2$
$\Rightarrow {{y}_{2}}=12$
From this we can find the slope
By substituting the above values in the formula of slope we will get
$\Rightarrow m=\dfrac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}$
By substituting the values in their respective positions or places we will get,
$\Rightarrow m=\dfrac{12-2}{5-3}$
By simplifying further, we will get,
$\Rightarrow m=\dfrac{10}{2}$
By simplifying further, we will get,
$\Rightarrow m=5$
Therefore, the slope of the given two points $\left( 3,2 \right),\ \left( 5,12 \right)$ is $5$.

Note: Students should know the basic formulas of two-dimensional coordinate geometry. Some of the important formulas by using slope are, if a line equation is $y=mx+c$ then the slope of the line is ”m”. if a line passes through the two points they are $\left( {{x}_{1}},{{y}_{1}} \right), \ \left( {{x}_{2}},{{y}_{2}} \right)$ then the line equation is $y-{{y}_{1}}=\dfrac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}\left( x-{{x}_{1}} \right)$ where $\dfrac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}$ is the slope of the line.