Writing a Linear Equation in Point-Slope Form

The slope and a point on a certain line are needed to write an equation in point-slope form. The slope can be found using two points on the line. That means this process begins by assuming that two points are given. After the slope is calculated, either point can be used to determine the equation. This method is illustrated using the following points that lie on a line. (-1,5) and (1,1) Three steps are needed to find the equation in point-slope form.

Calculate the Slope

First, the slope of the line can be calculated by substituting the given points into the Slope Formula. Note that if the slope is given, this step is not needed. If only the line is provided, select two points on the line whose coordinates are easy to identify.

m = y_2-y_1/x_2-x_1

SubstitutePoints

Substitute ( -1,5) & ( 1,1)

m=1- 5/1-( -1)

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Evaluate right-hand side

SubNeg

a-(- b)=a+b

m=1-5/1+1

AddSubTerms

Add and subtract terms

m=-4/2

CalcQuot

Calculate quotient

m=-2

In this case, the slope of the line that passes through the given points is - 2.

Select One Point on the Line

Next, one point on the line needs to be selected. Ideally, one of the points used in the previous step is chosen, but it can be any other point on the line whose coordinates are known. Out the given points, (1,1) will be used.

Substitute Values

Finally, once the slope and a point on the line are known, the equation can be written by substituting these values into the general equation in point-slope form. Here, m= -2 and (x_1,y_1)=( 1, 1). y-y_1&=m(x-x_1) &⇓ y- 1&= -2(x- 1)

Note that, unlike the slope-intercept form, an infinite number of equations written in point-slope form can represent the same line. In other words, when written in point-slope form, any line does not have only one unique equation.

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