Slope-Intercept and Point-Slope Forms of a Linear Equation

To determine the equation of a line, you may use two variations of the general form of a line. These formulas are:

  1. 1)  The Point-Slope Formula (y – y1) = m(x – x1)

  2. 2)  The Slope-Intercept Formula y = mx + b

As the names imply the form that you use is dependant on the information you are given to start with.

Example 1:

Solution

Step 1:


Find the equation of the line that has a slope of 3 and contains the point (2, -1).

Since the information given is a point and the slope, the point slope formula is used.

Substitute the given into the formula.

Sincem= 13 andP1 =(2,-1)thenx1 =2andy1 =-1.

y – y1 = m (x – x1)
y – (-1) = (x – 2)
y + 1 = (x – 2)
3 (y + 1) = 1(x – 2)
3y + 3 = x – 2
5=x–3y or x–3y=5

(This the standard formula of the line)

Calculate P2.

Select any value you with for x or y and substitute it into the equation found in step 1. For this example y will equal 2.

x – 3y = 5 x – 3(2) = 5 x–6=6
x = 11

Therefore P2 = (11, 2)

Step 3:

Verify.

When any two points of a line are substituted into the slope formula the slope of the line should be the answer. In this case, when P1 and P2 are substituted into the slope formula the answer should be 1/3.

SinceP1 =(2,-1)andP2 =(11,2)thenx1 =2,x2 =11,y1 =-1andy2 =2then:

Y−Y =1 21

X2 −X1 3

2 − (−1) = 1 11−2 3

3=1 93 1=1 33

(The slopes are alike so the equation and P2 are correct)