Question

How do you find the slope of a line given the points $\left( 2,2 \right)$ and $\left( 5,8 \right)$ on the line?

Answer 1

Here in this question we have to find the slope of the line passing through the given points. To find the slope of a line passing through points $\left( {{x}_{1}},{{y}_{1}} \right)$ and $\left( {{x}_{2}},{{y}_{2}} \right)$we will use the formula $\dfrac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}$. By substituting the values in the formula and simplifying the obtained equation we get the desired answer.

Complete step-by-step solution:
We have been given the points $\left( 2,2 \right)$ and $\left( 5,8 \right)$ on the line.
We have to find the slope of the line passing through the given points.
Now, we know that the slope of a line passing through the points $\left( {{x}_{1}},{{y}_{1}} \right)$ and $\left( {{x}_{2}},{{y}_{2}} \right)$ is given by the formula $\dfrac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}$.
Now, we have ${{x}_{1}}=2,{{y}_{1}}=2,{{x}_{2}}=5,{{y}_{2}}=8$
Now, substituting the values in the formula we will get
$\Rightarrow \dfrac{8-2}{5-2}$
Now, simplifying the above obtained equation we will get
$\begin{aligned} & \Rightarrow \dfrac{6}{3} \\\ & \Rightarrow 2 \\\ \end{aligned}$
Hence we get the slope of the line passing through the points $\left( 2,2 \right)$ and $\left( 5,8 \right)$ as 2.

Note: Alternatively we can first find the equation of the line passing through the given points by using the formula $\left( y-{{y}_{2}} \right)=\left( \dfrac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}} \right)\left( x-{{x}_{2}} \right)$. Now, we know that the equation of slope-intercept form of the line is given as $y=mx+c$, where m is the slope of the line and c is the y-intercept of the line. So by comparing the values we get the slope of the line.