Question

How do you find the slope of $( - 4,{\kern 1pt} {\kern 1pt} {\kern 1pt} - 1)$, $( - 2,{\kern 1pt} {\kern 1pt} {\kern 1pt} - 5)$?

Answer 1

We will get a straight line joining the given two points. The slope of the line is the change in the value of $y$ with respect to $x$. For a straight line, if two points $A({x_1},{\kern 1pt} {\kern 1pt} {\kern 1pt} {y_1})$ and $B({x_2},{\kern 1pt} {\kern 1pt} {\kern 1pt} {y_2})$ are situated on the line, then by using the slope formula we can calculate the slope (m) as, $m = \dfrac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}$.

Complete step by step solution:
We have to find the slope of the line passing through the points $( - 4,{\kern 1pt} {\kern 1pt} {\kern 1pt} - 1)$, $( - 2,{\kern 1pt} {\kern 1pt} {\kern 1pt} - 5)$.
As we already know two points on the line, we will use the slope formula to find the slope of the line.
The slope formula is given by $m = \dfrac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}$
where, $A({x_1},{\kern 1pt} {\kern 1pt} {\kern 1pt} {y_1})$ and $B({x_2},{\kern 1pt} {\kern 1pt} {\kern 1pt} {y_2})$ are the two points on the line
$m$ is the slope of the line
From the given points we can write,
${x_1} = - 4$, ${y_1} = - 1$, ${x_2} = - 2$ and ${y_2} = - 5$
Putting the values in the formula we get,
$\Rightarrow m = \dfrac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}} \\\ \Rightarrow m = \dfrac{{( - 5) - ( - 1)}}{{( - 2) - ( - 4)}} \\\ \Rightarrow m = \dfrac{{ - 5 + 1}}{{ - 2 + 4}} \\\ \Rightarrow m = \dfrac{{ - 4}}{2} = - 2 \\\$
Thus, the value of $m$ is $- 2$.

Hence, the slope of the line passing through $( - 4,{\kern 1pt} {\kern 1pt} {\kern 1pt} - 1)$ and $( - 2,{\kern 1pt} {\kern 1pt} {\kern 1pt} - 5)$ is $- 2$.

Note: For a line making obtuse angle with the x-axis, the slope is negative as the behavior of $y$ is opposite to that of $x$, i.e. the value of $y$ decreases for increase in the value of $x$ and the value of $y$ increases for decrease in the value of $x$. We can also find the slope of the line by first calculating the equation of the line passing through the given points and then using the slope-intercept formula $y = mx + c$, where $m$ is the slope of the line and $c$ is the y-intercept. The choice of the method depends on the information given in the question and the ease of solution.