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# Find the slope of the line passing through the following points: $\left( 3,-5 \right)$ and $\left( 1,2 \right)$?

Last updated date: 13th Jun 2024
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Hint: We start solving the problem by drawing the figure representing the given information and then assigning the variable for the slope of the given line. We then recall the definition of slope of the line passing through the points $\left( {{x}_{1}},{{y}_{1}} \right)$ and $\left( {{x}_{2}},{{y}_{2}} \right)$ as $\dfrac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}$. We then use this definition for the given points $\left( 3,-5 \right)$ and $\left( 1,2 \right)$. We then make the necessary calculations to get the required value of slope of the line.

According to the problem, we need to find the slope of the line passing through the following points: $\left( 3,-5 \right)$ and $\left( 1,2 \right)$.
Let us draw the figure representing the given information.

Let us recall the formula to find the slope passing through the points $\left( {{x}_{1}},{{y}_{1}} \right)$ and $\left( {{x}_{2}},{{y}_{2}} \right)$.
We know that the slope of the line passing through the points $\left( {{x}_{1}},{{y}_{1}} \right)$ and $\left( {{x}_{2}},{{y}_{2}} \right)$ is $\dfrac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}$.
Let us assume the slope of the line passing through the points $\left( 3,-5 \right)$ and $\left( 1,2 \right)$ be ‘m’.
So, we have $m=\dfrac{2-\left( -5 \right)}{1-3}$.
$\Rightarrow m=\dfrac{2+5}{-2}$.
$\Rightarrow m=\dfrac{7}{-2}$.
$\Rightarrow m=\dfrac{-7}{2}$.
So, we have found the slope of the line passing through the points $\left( 3,-5 \right)$ and $\left( 1,2 \right)$ as $\dfrac{-7}{2}$.

∴ The slope of the line passing through the points $\left( 3,-5 \right)$ and $\left( 1,2 \right)$ is $\dfrac{-7}{2}$.

Note: We can also solve this problem by using the formula of slope as $\dfrac{{{y}_{1}}-{{y}_{2}}}{{{x}_{1}}-{{x}_{2}}}$. We can also find the equation of the line first and then compare with the standard form to find the slope of the line. We can also find the angle made by the line with the line using the fact that the slope is tangent of the angle made by the line with x-axis. Similarly, we can expect problems to find the slope of the perpendicular to the given line.