Forms of the Equation of a Line

Example Problem 2

Find the equation of a line with a slope equal to 3 that passes through the point ( 1, 2 ).

Solution 2

Basically, you can use one of two approaches. The first approach is to use the point-slope formula and the second is to use the slope-intercept. To illustrate this, we will do it both ways.

( 1 ) Solve using the point-slope form

To use the point-slope form in determining the equation of the line, we simply plug in the values of m and the point
( x1, y 1). Therefore, the equation of the line would be,

   y - 2 = 3 ( x - 1 )  

( 2 ) Solve using the slope-intercept form

To use the slope-intercept form in determining the equation of the line, we simply plug in the values of m and the point
( x1, y 1) and then solve for b. Therefore,

   y = mx + b
 y = ( 3 )x + b
 ( 2 ) = ( 3 )( 1 ) + b
 2 - 3 = b
 b = -1
 

Once we have obtained b, all we have to do is plug the values of m and b into the slope-intercept formula and we're done.

   y = mx + b
 y = ( 3 )x + ( -1 )
 y = 3x - 1
 

( 3 ) Prove that the two equations are equal

Now that we have solved the problem using both the
point-slope and the slope-intercept formula's, we would like to prove that both equations are, in fact, equal. To do so, we will simply convert the equation in point-slope form into slope-intercept form.

   y - 2 = 3 ( x - 1 )
 y - 2 = 3x - 3
 y = 3x - 3 + 2
 y = 3x - 1
 

As you can see, both approaches will give you the correct equation of the line. The only difference is which form it is in.

 

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