Question

How do you write the equation in point slope from given $\left( -1,7 \right)$ and $\left( 8,-2 \right)$?

Answer 1

Hint: In this problem we need to write the equation of the line which passes through the given point in the point slope form. For this we need to calculate the slope of the line which passes through the given equation. We know that the slope of the line passes through the points $\left( {{x}_{1}},{{y}_{1}} \right)$ and $\left( {{x}_{2}},{{y}_{2}} \right)$ is given by $m=\dfrac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}$. First, we will compare the given two points with $\left( {{x}_{1}},{{y}_{1}} \right)$ and $\left( {{x}_{2}},{{y}_{2}} \right)$ to calculate the value of slope. After calculating the slope of the line, we can use the point slope formula which is $\left( y-{{y}_{1}} \right)=m\left( x-{{x}_{1}} \right)$ to get the equation of the line.

Complete step-by-step solution:
Given points are $\left( -1,7 \right)$ and $\left( 8,-2 \right)$.
Comparing the above points with $\left( {{x}_{1}},{{y}_{1}} \right)$ and $\left( {{x}_{2}},{{y}_{2}} \right)$, then we will get
${{x}_{1}}=-1$, ${{y}_{1}}=7$, ${{x}_{2}}=8$, ${{y}_{2}}=-2$.
Now the slope of the line which is passing through the points $\left( -1,7 \right)$ and $\left( 8,-2 \right)$ is given by
$ \Rightarrow m=\dfrac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}$
Substituting the values, we have in the above equation, then we will get
$\begin{align}
  & \Rightarrow m=\dfrac{-2-7}{8-\left( -1 \right)} \\
 & \Rightarrow m=\dfrac{-9}{8+1} \\
 & \Rightarrow m=-\dfrac{9}{9} \\
 & \Rightarrow m=-1 \\
\end{align}$
Now the equation of the line which passes through the points $\left( -1,7 \right)$ and $\left( 8,-2 \right)$ in slope point form is given by
$\begin{align}
  & \Rightarrow y-{{y}_{1}}=m\left( x-{{x}_{1}} \right) \\
 & \Rightarrow y-7=-1\left( x-\left( -1 \right) \right) \\
 & \Rightarrow y-7=-1\left( x+1 \right) \\
\end{align}$
Hence the equation of the line which passes through the given points $\left( -1,7 \right)$ and $\left( 8,-2 \right)$ is $y-7=-1\left( x+1 \right)$.

Note: In this problem they have asked to calculate the equation of the line in slope point form so we have substituted the value in the slope point form. We can also simplify the obtained equation to convert the line equation into standard from. We can write the equation of the line in standard from which passes through the given points $\left( -1,7 \right)$ and $\left( 8,-2 \right)$ as
$\begin{align}
  & \Rightarrow y-7=-x-1 \\
 & \Rightarrow x+y-7+1=0 \\
 & \Rightarrow x+y-6=0 \\
\end{align}$