Writing Equations Using Parallel Lines

Writing Equations Using Parallel Lines

Parallel Lines have the same slope!

Defining Parallel Lines

➽ Parallel Lines are two lines that remain equidistant from one another forever.  In other words, they don’t get any closer or any farther apart from each other forever.  

➽ Parallel Lines will never intersect.

➽ Parallel Lines have the same slope.  Each line will have the same rise over run from any point as the line parallel to it does.   This is an important point for us  -   Parallel lines have the same slope!

Writing Equations using Parallel Lines

➽  When asked to write an equation of a line that is parallel to another line, we need the slope of that line.  We don’t need any other information from the parallel line.  We find the slope by rewriting the equation in slope-intercept form.

What to Do when Asked to Write the Equation of a Line Parallel to another

To write the equation of a line parallel to a line.

*Remember the line given will be written as the equation of the line.

*We need the slope of that line. 

*We don’t need the y-intercept or any other information from that equation.

*We find the slope of that line by rewriting the equation in slope-intercept form. 

*The slope is the coefficient of x.

Writing the Equation of a Line parallel to another line.

To write the equation of a line parallel to the given line.

*After finding the slope

*Use the slope (Because Parallel Lines have the same slope) and the point that our line passes through along with point-slope form to write the equation.

*Rewrite the equation in the form the directions ask for.

Example

Example:  A line passes through (-4, 5) and is parallel to 4x - 2y = 10.  Write the equation of the line in slope-intercept form.

We must first rewrite the given equation in slope-intercept form.

Remember that the given equation represents a line.

See the next slide for the work!

Example Continued

4x - 2y  = 10

 4x - 2y - 4x = 10 - 4x Subtract 4x from each side to isolate -2y

            -2y    =   -4x + 10

            -2y    =   -4x + 10 Divide each side by -2

             -2            -2    -2

               y   =     2x - 5

This is equation of the line in slope intercept form

The slope is 2  

We will use the slope of 2 for our line (Parallel Lines have the same slope) 

  and the point (-4, 5).

All we need from this equation (since it is parallel to our line) is the slope.

After finding the slope - Write the Equation

We will use point-slope form to write the equation.

         y - y₁ = m(x - x₁) We will substitute for the slope, m and the point  (x₁, y₁)

         y - 5 = 2 (x - -4) Substitute in for the point and the slope

         y - 5 = 2 (x + 4) Double negatives mean to change to a (+) 

This is the equation in point-slope form

Rewriting the Equation in Slope-Intercept Form

  y - 5 = 2x + 8 Distributive Property

  y - 5 + 5 =  2x + 8 + 5 Add 5 to each side

            y =  2x + 13

This is the equation in slope intercept form. 

        

Rewriting the Equation in Standard Form

    y = 2x + 13

   y - 2x = 2x + 13 - 2x Subtract 2x on each side

 -2x + y = 13

-1[-2x + y = 13] Multiply the equation by -1     2x - y = -13

This is the equation in standard form

Multiple Choice

A line is represented by 4x - 8y = 16. What is the slope of the line? [Hint: Turn into Slope Intercept Form]

• m = 2

• m = -2

• m = 1/2

• m = -1/2

Multiple Choice

Write the equation in point-slope form of the line that is parallel to 4x - 8y = 16 and passes through (-1, 6).

• y + 1 = 4(x - 6)

• y - 6= 1/2 (x + 1)

• y - 6 = -1/2 (x - 1)

• y - 6 = -1/2 (x + 1)

Multiple Choice

Rewrite 

 $y\ -\ 6\ =\ \frac{1}{2}\left(x\ +\ 1\right)$  in slope intercept form. 

• x + 2y  = 13

•  $y\ =\ \frac{1}{2}x\ -\ \frac{11}{2}$  

•  $y\ =\ \frac{1}{2}x\ +\ \frac{13}{2}$  

• y  = x +13