Forms of the Equation of a Line

Definitions

The Standard Form of the Equation of a Line

Ax + By = C

where A, B, and C are real numbers



 

The Point-Slope Form of the Equation of a Line

The equation of a line with slope m and passing through
the point ( x1, y 1) is given by,

y - y 1 = m ( x - x1 )

where m is the slope and ( x1, y 1) is the point given


 

The Slope-Intercept Form of the Equation of a Line

The equation of a line with slope m and
y-intercept ( 0, b ) is given by,

y = mx + b

where m is the slope and ( 0, b ) is the y-intercept


 

Example Problem 1

Find the equation of the line that passes through the points
( -3, 4 ) and ( 1, 7 ).

Solution 1

The first step is to determine the slope of the line. In order to determine the slope of the line, we must use the formula
m = ( y2 - y1 ) / ( x2 - x1 ). This gives us,

   m = ( y2 - y1 ) / ( x2 - x1 )
 m = ( 7 - 4 ) / ( 1 - ( -3 ) )
 m = ( 7 - 4 ) / ( 1 + 3 )
 m = 3/4
 

Therefore, the slope of the line is equal to 3/4.

Now, the next step is to apply the point-slope formula. To do so, we must choose one of the points, ( x1, y1 ), and insert it and the slope into the formula which will give us,

   y - y1 = m ( x - x1 )
 y - 4 = ( 3/4 )( x - ( -3 ) )
 y - 4 = ( 3/4 )( x + 3 )
 

Finally, now that we have the equation of the line in
point-slope form, we will want to convert the equation into
slope-intercept form. ( This will allow us to determine the
 y-intercept directly from the formula. )

   y - 4 = ( 3/4 )( x + 3 )
 y - 4 = ( 3/4 )x 9/4
 y = ( 3/4 )x 9/4 + 4
 y = ( 3/4 )x 9/4 + 16/4
 y = ( 3/4 )x + 25/4
 

As you can see, when x = 0, y = 25/4. We now have the equation of the line in slope-intercept form and the
  y-intercept is at the point ( 0, 25/4 ).

Note: If the question does not specify a specific form for the equation of the line then an equation in either point-slope or slope-intercept form would be a correct answer.

 

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