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 = m₁x + c₁ and y = m₂x + c₂ are parallel if m₁ = m₂.

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 = 7x + 10.

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

Example 2: Write the equation of a line parallel to y = 7x + 10 which passes from (1, 1).

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

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

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

Hence, the required equation is y = 7x – 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 = m₁x + c₁ and y = m₂x + c₂ are perpendicular if m₁m₂ = -1 or m_{1 =}

Example 3: Find the equation of a line perpendicular to y = 2x + 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 – $\frac{1}{m}$ = – $\frac{1}{2}$ .

Equation of line perpendicular to given line is y= – $\frac{1}{m}$ x + k = – $\frac{1}{2}$ x+k, where k is any real number.

Example 4: Find the equation of a line perpendicular to y = 3x + 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 – $\frac{1}{m}$ = – $\frac{1}{3}$ .

Equation of line passing from (1, 2) and slope – $\frac{1}{3}$ is (y – 2) = – $\frac{1}{3}$ (x-1)

3(y – 2) = -(x – 1)

3y – 6 = –x + 1

x + 3y = 7

Note:

(1) If two lines are parallel,then their slopes are equal.

(2) If two lines are perpendicular,then product of their slopes is -1.

To find the slope of a line perpendicular to given line,We take the reciprocal and change its sign.

Hence,slope of a line perpendicular to a given line with slope m is -1/m.

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 m₁ = -1.

The slope of second line, y = (-1)x + 100 is m₂ = -1.

m₁ = m₂ = -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 m₁ = -1.

The slope of second line, y = (1)x – 100 is m₂ = 1.

m₁ m₂ = -1, so the given lines are perpendicular.

Check Point

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

Answer Key

The required parallel line is y = 5x + 10.
The required perpendicular line is y = – $\frac{1}{5}$ x + $\frac{26}{5}$ or x + 5y = 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 = 7x + 51 is y = 7x + k, where k is any real
The equation of the line perpendicular to the line y = 7x + 51 is y = – $\frac{1}{7}$ x + k , where k is any real

Personalized Online Tutoring

eTutorWorld offers affordable one-on-one live tutoring over the web for Grades 2-12, Test Prep help for Standardized tests like SCAT, CogAT, SSAT, SAT, ACT, ISEE and AP. You may schedule online tutoring lessons at your personal scheduled times, all with a Money-Back Guarantee. The first one-on-one online tutoring lesson is always FREE, no purchase obligation, no credit card required.

For answers/solutions to any question or to learn concepts, take a FREE Demo Session.

Schedule a Free Session

No credit card required, no obligation to purchase.
Just schedule a FREE Sessions to meet a tutor and get help on any topic you want!

Learn more about Scientific Method and other important topics with 7th Grade Science Tutoring at eTutorWorld. Our expert science tutors break down the topics through interactive one-to-one sessions. We also offer the advantage of customized lesson plans, flexible schedules and convenience of learning from home.

Pricing for Online Tutoring

Tutoring Package	Validity	Grade (1-12), College
5 sessions	1 Month	$124
1 session	1 Month	$25
10 sessions	3 months	$239
15 sessions	3 months	$354
20 sessions	4 months	$449
50 sessions	6 months	$1049
100 sessions	12 months	$2049

Buy Now