Finding the Equation of a Line Given 2 Points

To find the equation of a line given 2 points, follow this 2 step process:

Step 1:  Find the slope using the formula

                    y₂ - y₁
        m  =                    
                    x₂ - x₁

Step 2:  Find the equation of the line using the point-slope formula

        y - y₁  =  m(x - x₁)

Then multiply out, solve for y and simplify.

Example 1

Find the equation of the line through the points (2,5) and (4,1)

Solution

Step 1:  We find

                    1 - 5
        m  =                 
                    4 - 2

                  -4
         =               =  -2
                  2

Step 2:  Use the point-slope formula

        y - 5  =  -2(x - 2)

Multiply through to get

        y - 5  =  -2x + 4

Now add 5 to both sides to get

        y  =  -2x +9

Example 2

Find the equation of the line through the points (-3,4) and (-2,-5)

Solution

Step 1:  We find

                    -5 - 4
        m  =                  
                    -3 - 2

                  -9             9
         =               =            
                  -5             5

Step 2:  Use the point-slope formula

        y - 4  =  9/5(x - (-3))

Multiply through to get

        y - 4  =  9/5 x + 27/5

Now add 4 to both sides to get

        y  =   9/5 x + 27/5 + 4

we need to add the fractions:

        27        4
              +             
         5         1

          27        20
    =          +             
           5         5

          47  
    =            
           5

We now have the equation of he line:

    y  =   9/5 x + 47/5

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