The first option could be produced with the points (-1, -2) and (4, -3)
Step-by-step explanation:
Step 1:
Assume [tex](x_{1}, y_{1}) = (-1,-2)[/tex] and [tex](x_{2}, y_{2}) = (4,-3)[/tex]. So [tex]x_{1}=-1, y_{1}=-2[/tex] and [tex]x_{2}=4, y_{2}=-3[/tex] .
The point-slope form is [tex]\left(y-y_{1}\right)=m\left(x-x_{1}\right)[/tex].
Step 2:
First, we need to determine the value of the slope, m to substitute in the equation.
The slope is given by [tex]m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}[/tex].
[tex]m=\frac{-3-(-2)}{4-(-1)} = \frac{-3+2}{4+1}= -\frac{1}{5}.[/tex]
Step 3:
Substituting m's value in the equation, we get
For [tex]x_{1}=-1, y_{1}=-2[/tex], [tex]\left(y-y_{1}\right)=m\left(x-x_{1}\right) = \left(y-(-2)}\right)=-\frac{1}{5} \left(x-(-1)}\right)[/tex] [tex]=\left(y+2}\right)=-\frac{1}{5} \left(x+1}\right)[/tex].
For [tex]x_{2}=4, y_{2}=-3[/tex], [tex]\left(y-y_{1}\right)=m\left(x-x_{1}\right) = \left(y-(-3)}\right)=-\frac{1}{5} \left(x-4}\right)[/tex][tex]=\left(y+3}\right)=-\frac{1}{5} \left(x-4}\right)[/tex].
The first option matches the calculated value [tex]\left(y+2}\right)=-\frac{1}{5} \left(x+1}\right)[/tex].
