Point-Slope Equation

The point-slope equation is a way of describing the equation of a line. The point-slope form is ideal if you are given the slope and only one point, or if you are given two points and do not know what the $y$-intercept is. Given a slope, $m$, and a point $(x_{1}, y_{1})$, the point-slope equation is: 

$\displaystyle y-y_{1}=m(x-x_{1})$ 

Verify Point-Slope Form is Equivalent to Slope-Intercept Form

To show that these two equations are equivalent, choose a generic point $(x_{1}, y_{1})$. Plug in the generic point into the equation $y=mx+b$.  The equation is now, $y_{1}=mx_{1}+b$, giving us the ordered pair,$(x_{1}, mx_{1}+b)$. Then plug this point into the point-slope equation and solve for $y$ to get:

$\displaystyle y-(mx_{1}+b)=m(x-x_{1})$

Distribute the negative sign through and distribute $m$ through $(x-x_{1})$:

$\displaystyle y-mx_{1}-b=mx-mx_{1}$

Add $mx_{1}$ to both sides:

$\displaystyle y-mx_{1}+mx_{1}-b=mx-mx_{1}+mx_{1}$

Combine like terms:

$\displaystyle y-b=mx$

Add $b$ to both sides:

$\displaystyle y-b+b=mx+b$

Combine like terms:

$\displaystyle y=mx+b$

Therefore, the two equations are equivalent and either one can express an equation of a line depending on what information is given in the problem or what type of equation is requested in the problem.

Example: Write the equation of a line in point-slope form, given a point $(2,1)$ and slope $-4$, and convert to slope-intercept form  

Write the equation of the line in point-slope form:  

$\displaystyle y-1=-4(x-2)$

To switch this equation into slope-intercept form, solve the equation for $y$:

$\displaystyle y-1=-4(x-2)$  

Distribute $-4$:

$\displaystyle y-1=-4x+8$  

Add $1$ to both sides:

$\displaystyle y=-4x+9$

The equation has the same meaning whichever form it is in, and produces the same graph.

Line graph

Graph of the line $y-1=-4(x-2)$, through the point $(2,1)$ with slope of $-4$, as well as the slope-intercept form, $y=-4x+9$.

Example: Write the equation of a line in point-slope form, given point $(-3,6)$ and point $(1,2)$, and convert to slope-intercept form

Since we have two points, but no slope, we must first find the slope: 

$\displaystyle m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}$ 

Substituting the values of the points: 

$\displaystyle \begin{aligned} m&=\frac{-2-6}{1-(-3)}\\&=\frac{-8}{4}\\&=-2 \end{aligned}$ 

Now choose either of the two points, such as $(-3,6)$. Plug this point and the calculated slope into the point-slope equation to get:

 $\displaystyle y-6=-2[x-(-3)]$

Be careful if one of the coordinates is a negative.  Distributing the negative sign through the parentheses, the final equation is: 

$\displaystyle y-6=-2(x+3)$

If you chose the other point, the equation would be: $y+2=-2(x-1)$ and either answer is correct. 

Next distribute $-2$:

$\displaystyle y-6=-2x-6$   

Add $6$ to both sides:

$\displaystyle y=-2x$

 Again, the two forms of the equations are equivalent to each other and produce the same line.  The only difference is the form that they are written in.