Question

How do you find the slope of the line passing through each pair of points. $\left( -3,-1 \right),\left( -1,5 \right)?$

Answer 1

Hint: Slope is the change in the $y$-value divided by the change in the $x$-values.
Slope $=\dfrac{\text{Rate of change }my}{\text{Rate of change }mx}=\dfrac{rise}{sun}$
$m=\dfrac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}$
It is often expressed as rise over run. Identify $\left( {{x}_{1}},{{y}_{1}} \right)$ and $\left( {{x}_{2}},{{y}_{2}} \right)$ from given question.

Complete step by step solution:
As per the given problem, $\left( -3,-1 \right)$ and $\left( -1,5 \right)$ we the two points passing through the slope.
So, here you have to identify $\left( {{x}_{1}},{{y}_{1}} \right)$ and $\left( {{x}_{2}},{{y}_{2}} \right)$ from the given points.
Therefore, $\left( -3,-1 \right)$ and $\left( -1,5 \right)$
${{x}_{1}}=-3$
${{x}_{2}}=-1$
${{y}_{1}}=-1$
${{y}_{2}}=5$
Because points are written in $\left( {{x}_{1}},{{y}_{1}} \right)$ and $\left( {{x}_{2}},{{y}_{2}} \right)$ form.
Here, formula for finding the slope of the line passing two points is.
Slope of line $\left( m \right)=\dfrac{Rise}{Run}$
$=\dfrac{\text{Rate of change in value of }y}{\text{Rate of change in value of }x}$
$=\dfrac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}$
$=\dfrac{5+1}{-1+3}$
$=\dfrac{6}{2}$
Therefore,
Slope of the line $\left( m \right)=3$
Hence slope of line passing through each pair of points $\left( -3,-1 \right)$and $\left( -1,5 \right)$ is $3.$

Additional Information:
The slope is a number that tells how much $'y'$ changes when $'x'$ changes. For example: a slope of $5$ means that for each change in $x$ of $1$ unit (for example between $6$ and $7)$ the corresponding $y$ changes of $5$units. This is for a positive slope, so that your value of $y$ is getting bigger.
The negative slope is the opposite. It tells you of how much $y$ decreases for each in orase of $1$ unit in $x.$ A slope of $-5$ tells you that the value of $y$-decreases of $5.$ units in the $1$unit interval of $x.$
Remember the slope of line is defined as
$Slope=\dfrac{Rise}{Run}$
And the rise of the horizontal line is zero since it is neither increasing nor decreasing. Therefore the slope of the line is zero.

Note:
Apply the formula of slope of line $m=\dfrac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}$
Where, ${{x}_{2}},{{x}_{1}}$ are the $x$ component of a given pair. And ${{y}_{1}},{{y}_{2}}$ are the $y$ components of a given pair.
Always remember that as in the formula $'-{{y}_{1}}'$ is given and if the value of $'{{y}_{1}}'$ is $'-1'$
Therefore it will become $+1$
$'-'\times '-'='+'$