Question

Write an equation in point-slope form for the line that passes through the points (f, g) and (h,j).

Answer 1

Point-Slope Form

Point-slope form is a form of a linear equation: $y-y_1=m(x-x_1)$

• $(x_1,y_1)$ is a point that falls on the line
• m is the slope of the line

To write an equation in point-slope form:

1. Calculate the slope of the line by solving for m
2. Plug m into the general equation
3. Plug a point that falls on the line in the general equation as $(x_1,y_1)$

Solving the Question

We're given:

• The line passes through the points (f,g) and (h,j)

Solve for the slope (m):

$y-y_1=m(x-x_1)$

⇒ Plug in the point (f,g) as (x,y):

$g-y_1=m(f-x_1)$

⇒ Plug in the point (h,j) as (x₁,y₁):

$g-j=m(f-h)$

⇒ Isolate m by dividing both sides by (f-h):

$\dfrac{g-j}{f-h}=m$

⇒ Therefore, the slope of the line is $\dfrac{g-j}{f-h}$.

Plug the slope into the general equation:

$y-y_1=\dfrac{g-j}{f-h}(x-x_1)$

Plug one of the points, (f,g) or (h,j) into the equation as (x₁,y₁):

$y-j=\dfrac{g-j}{f-h}(x-h)$

Answer

There can be multiple answers for this question, depending on what we consider to be (x,y) and what we consider to be (x₁,y₁). This is one of the possible answers, for (f,g) is (x,y) and (h,j) is (x₁,y₁):

$y-j=\dfrac{g-j}{f-h}(x-h)$