Equation of a Line Parallel to a Line

We will learn how to find the equation of a line parallel to a line.

Prove that the equation of a line parallel to a given line ax + by + λ = 0, where λ is a constant.

Let, ax + by + c = 0 (b ≠ 0) be the equation of the given straight line.

Now, convert the equation ax + by + c = 0 to its slope-intercept form.

ax + by+ c = 0

⇒ by = - ax - c

Dividing both sides by b, [b ≠ 0] we get,      

y =  -\(\frac{a}{b}\) x - \(\frac{c}{b}\), which is the slope-intercept form.

Now comparing the above equation to slope-intercept form (y = mx + b) we get,

The slope of the line ax + by + c = 0 is (- \(\frac{a}{b}\)).

Since the required line is parallel to the given line, the slope of the required line is also (- \(\frac{a}{b}\)).

Let k (an arbitrary constant) be the intercept of the required straight line. Then the equation of the straight line is

y = - \(\frac{a}{b}\) x + k

by = - ax + bk        

ax +  by = λ, Where λ = bk = another arbitrary constant.

Note: (i) Assigning different values to λ in ax + by = λ we shall get different straight lines each of which is parallel to the line ax + by + c = 0. Thus, we can have a family of straight lines parallel to a given line.

(ii) To write a line parallel to a given line we keep the expression containing x and y same and simply replace the given constant by a new constant λ. The value of λ can be determined by some given condition.

To get it more clear let us compare the equation ax + by = λ with equation ax + by + c = 0. It follows that to write the equation of a line parallel to a given straight line we simply need to replace the given constant by an arbitrary constant, the terms with x and y remain unaltered. For example, the equation of a straight line parallel to the straight line 7x - 5y + 9 = 0 is 7x - 5y + λ = 0 where λ is an arbitrary constant.

Solved examples to find the equations of straight lines parallel to a given line:

1.  Find the equation of the straight line which is parallel to 5x - 7y = 0 and passing through the point (2, - 3).

Solution:    

The equation of any straight line parallel to the line 5x - 7y = 0 is 5x - 7y + λ = 0 …………… (i)  [Where λ is an arbitrary constant].

If the line (i) passes through the point (2, - 3) then we shall have,

5 ∙ 2 - 7 ∙ (-3) + λ = 0

10 + 21 + λ = 0

31 + λ = 0

λ = -31

Therefore, the equation of the required straight line is 5x - 7y - 31 = 0.


2. Find the equation of the straight line passing through the point (5, - 6) and parallel to the straight line 3x - 2y + 10 = 0.

Solution:

The equation of any straight line parallel to the line 3x - 2y + 10 = 0 is 3x - 2y + k = 0 …………… (i) [Where k is an arbitrary constant].

According to the problem, the line (i) passes through the point (5, - 6) then we shall have,

3 ∙ 5 - 2 ∙ (-6) + k = 0

15 + 21 + k = 0

36 + k = 0

k = -36

Therefore, the equation of the required straight line is 3x - 2y - 36 = 0.

 The Straight Line





11 and 12 Grade Math 

From Equation of a Line Parallel to a Line to HOME PAGE


New! Comments

Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question.



Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.



Share this page: What’s this?

Recent Articles

  1. 4th Grade Mental Math on Roman Numerals | Roman Numerals Quiz

    Feb 23, 24 03:55 PM

    In 4th grade mental math on numbers, students can practice different questions on write the Hindu-Arabic numerals, write the Roman Numerals, comparison of roman numerals, addition of roman numerals.

    Read More

  2. 4th Grade Mental Math on Numbers | Mental Math 4th Grade with Answers

    Feb 23, 24 02:24 PM

    4th Grade Mental Math on Numbers
    In 4th grade mental math on numbers, students can practice different questions on numbers in figures, number name, place value, face value, comparison of number and formation of greatest and smallest…

    Read More

  3. Roman Numerals | System of Numbers | Symbol of Roman Numerals |Numbers

    Feb 23, 24 01:28 PM

    List of Roman Numerals Chart
    How to read and write roman numerals? Hundreds of year ago, the Romans had a system of numbers which had only seven symbols. Each symbol had a different value and there was no symbol for 0. The symbol…

    Read More

  4. Worksheet on Roman Numerals |Roman Numerals|Symbols for Roman Numerals

    Feb 22, 24 04:15 PM

    Roman Numbers Table
    Practice the worksheet on roman numerals or numbers. This sheet will encourage the students to practice about the symbols for roman numerals and their values. Write the number for the following: (a) V…

    Read More

  5. Roman Symbols | What are Roman Numbers? | Roman Numeration System

    Feb 22, 24 02:30 PM

    Roman Numbers
    Do we know from where Roman symbols came? In Rome, people wanted to use their own symbols to express various numbers. These symbols, used by Romans, are known as Roman symbols, Romans used only seven…

    Read More