# Parallel and Perpendicular Lines in Graphs of Linear Equations

A **Linear equation** is an equation in which the highest exponent of the variable present in the equation is one. When we draw the **graph of the linear equation**, it **forms a straight** **line.**

If any two lines in the plane are drawn, they are either parallel or intersecting.

### How do you know if a line is parallel?

**Two lines are parallel if their slopes are equal**.

Hence the lines *y* = *m*_{1}*x *+ *c*_{1} and* y* = *m*_{2}*x *+ *c*_{2} are parallel if *m*_{1} = *m*_{2}.

In fact, **two parallel lines differ by a constant**.

Hence if the equation of line is *y* = *mx* + *c*, then, equation of a line parallel to it is *y* = *mx* + *k*, where *k* is a constant. To find the particular line, we require a unique value of *k*. For this additional condition is given.

** Example 1: **Write the equation of a line parallel to

*y*= 7

*x*+ 10.

Using the above concept, the equation of a line parallel to above line is *y* = 7*x* + *k*, where *k* is any real number.

** Example 2: **Write the equation of a line parallel to

*y*= 7

*x*+ 10 which passes from (1, 1).

The equation of a required line parallel to above line is* y* = 7*x* + *k*, where *k* is any real number.

Since the above line passes from (1, 1), hence it satisfies the required equation* y* = 7*x* + *k*.

Hence, 1 = 7(1) + *k* and *k* = 1 – 7 = -6.

Hence, the required equation is *y* = 7*x* – 6.

### How do you know if a line is perpendicular?

**Two lines are perpendicular if the product of their slopes is -1**.

Hence the lines *y* = *m*_{1}*x *+ *c*_{1} and* y* = *m*_{2}*x *+ *c*_{2} are perpendicular if *m*_{1}*m*_{2} = -1 or *m*_{1 = }

** Example 3:** Find the equation of a line perpendicular to

*y*= 2

*x*+ 5.

Comparing with the slope intercept form, *y* = *mx* + *c*, the slope of given line is *m *= 2. Hence, the slope of a line perpendicular to given line is –= – .

Equation of line perpendicular to given line is *y= – **x* + *k = – **x+k*, where *k* is any real number.

** Example 4: **Find the equation of a line perpendicular to

*y*= 3

*x*+ 5 passing from (1, 2).

As, done in the previous example slope of given line is *m* = 3.

Hence, the slope of a line perpendicular to given line is –* = –*.

Equation of line passing from (1, 2) and slope – is (*y* – 2) = –* (x-1)*

3(*y* – 2) = -(*x* – 1)

3*y* – 6 = –*x* + 1

*x* + 3*y* = 7

** Example 5:** Check whether the lines

*x*+

*y*= 10 and

*x*+

*y*= 100 are parallel or perpendicular.

Comparing with the slope intercept form,* y* = *mx* + *c*

The slope of first line, *y* = (-1)*x* + 10 is *m*_{1} = -1.

The slope of second line, *y* = (-1)*x* + 100 is* m*_{2} = -1. ** **

*m*_{1} = *m*_{2} = -1. Hence the given lines are parallel.

** Example 6:** Check whether the lines

*x*+

*y*= 10 and

*x*–

*y*= 100 are parallel or perpendicular.

Comparing with the slope intercept form,* y* = *mx* + *c*

The slope of first line, *y* = (-1)*x* + 10 is *m*_{1} = -1.

The slope of second line, *y* = (1)*x* – 100 is *m*_{2} = 1.

*m*_{1}* m*_{2} = -1, so the given lines are perpendicular.

**Check Point**

- Find the equation of the line passing through (-1, 5) & parallel to the line
*y*= 5*x*+ 1. - Find the equation of the line passing through (-1, 5) & perpendicular to the line
*y*= 5*x*+ 1. - Check whether the lines, 3
*x*+*y*= 15 and 21*x*+ 7*y*= 28 are parallel or perpendicular. - Check whether the lines 3
*x*+*y*= 15 and*x*– 3*y*= 28 are parallel or perpendicular. - Find the equation of the line parallel to the line,
*y*= 7*x*+ 51. - Find the equation of the line perpendicular to the line
*y*= 7*x*+ 51.

##### Answer Key

- The required parallel line is
*y*= 5*x*+ 10. - The required perpendicular line is y = –
*x +*or*x*+ 5*y*= 26. - Since, slopes of the two lines are equal which is -3. Hence the given lines are parallel.
- Since, product of the slopes of the two lines is -1. Hence the given lines are perpendicular.
- The equation of the line parallel to the line
*y*= 7*x*+ 51 is*y*= 7*x*+*k*, where*k*is any real - The equation of the line perpendicular to the line
*y*= 7*x*+ 51 is*y = – x + k*, where*k*is any real

*No credit card is required, nor are you under any obligation to make a purchase. Just schedule the FREE TRIAL lesson to meet a tutor & get help on any topic you want!*

#### Find more about our Personalized Online Tutoring Learning Packages.

+1-269-763-4602

+1-269-763-5024

Quick Links

Follow Us

#### Give Your Child The eTutorWorld Advantage

Research has proven that personal online tutoring not just cements school learning, it helps build student confidence. Come to eTutorWorld for Expert Tutors and best K-12 Online Tutoring Services in the comfort and safety of your home at an affordable cost. Find a tutor online for Grade 3-12 Math, Science and English subjects and AP, SAT, SSAT and SCAT Test Prep help and test practice. Get free printable math and science worksheets in pdf format and SCAT Practice Tests. Sign up for a Free Trial Lesson Today!

© 2018 eTutorWorld - Online Tutoring and Test Prep | All rights reserved