Here we will try to find out the number of possible solutions of any linear equation.

Now we consider each of the above cases separately and understand them with examples.

## Linear Equations With one Solution

*Example 1:* Consider the equation 7*x* – 35 = 0.

*Example 1:*

On solving we have 7*x* = 35 or *x* = 5. The above linear equation is only true if *x* = 5 and hence the given **linear equation has only one solution** i.e. *x* = 5.

*Example 2:* Consider the equation 9(*x* – 1) – 35 = 8*x* + 37.

On solving we have 9*x* – 9 – 35 = 8*x *+ 37.

Collect the like terms on both sides by transferring them, we have

9x – 8x = 37 + 35 + 9 = 80 which gives *x* = 80.

The above linear equation is only true if *x* = 80

Hence, the given **linear equation has only one solution** i.e.* x* = 80.

From the above examples, we see that the variable *x* does not disappear after solving & we say that the linear equation will have **one solution** if it is satisfied by exactly one value of the variable.

** Example 1: **Consider the equation 7

*x*– 35 = 5

*x*+ 2

*x*– 27.

On solving we have 7*x* – 35 = 7x – 27

Subtracting 7*x* from both sides. 7*x* – 7*x* – 35 = 7*x* – 7*x* – 27

we have -35 = -27 which is a false statement since it can’t be true for any value of the variable *x.*

Hence, the given **linear equation has zero solution** or the number of solutions is zero.

** Example 2:** Consider the equation 3(

*x*+ 9) + 21

*x*= 24

*x*+ 9.

On solving we have 3*x* + 27 + 21 *x *= 24*x* + 9 or 24 *x* + 27 = 24*x* + 9

Subtracting 24*x* form both sides, 24*x* – 24*x* + 27 = 24*x* – 24*x* + 9.

We have 27 = 9, which is a false statement since it can’t be true for any value of the variable *x.*

Hence, the given **linear equation has no solution** or the **number of solutions is zero**.

From the above examples, we can see that the variable x disappears / gets eliminated & hence we say that the linear equation will have **Zero or no solution** if it can’t be satisfied by any value of the variable or there does not exist any value of the variable which makes the given equation a true statement.

Linear equations with infinite solutions

*Example 1:* Consider the equation 25*x *– 35 = 5 (5*x *+ 4) – 55.

On solving we have 25*x *– 35 = 25*x* + 20 – 55 or 25*x* – 35 = 25*x* – 35.

Subtracting 25*x* from both sides, 25*x *– 25*x *– 35 = 25*x *– 25*x* – 35

We have -35 = -35, which is a True statement & it will be true for any value of the variable *x*.

Hence the given **linear equation has Infinite solutions** or the number of solutions is infinite.

*Example 2:* Consider the equation 15 (*x* + 9) = 24* x* + 9 – (9*x* – 126)

Solving we have 15*x* + 144 = 24 *x* + 9 – 9*x *+ 126 or 15 *x* + 144 = 15*x* + 144.

Subtracting 15*x* from both sides. 15*x* – 15*x* +144 = 15*x* – 15*x* + 144

We have 144 = 144, which is a True statement & it will be true for any value of the variable *x*.

Hence the given **linear equation has Infinite solutions** or the number of solutions is infinite.

From the above examples we can say that, the linear equation will have **infinite solutions** if it is satisfied by any value of the variable or every value of the variable makes the given equation a true statement.

**CHECK POINT**

Solve the following linear equations & identify whether the given linear equations have one , zero or infinite solutions.

- 17
*x –*75 = 6 + 14*x*. - 3
*x*– 105*=*4(*x*– 20) – 1(*x*+ 5). - 10
*x*+ 2*7 =*2(5*x*+ 99). - 7
*x*– 33 + 75 = 6(*x*+ 7) + x. - 24
*x*+ 60*=*4 (*x*– 25). - 13
*x*+ 10 – 4x*=*4(*x*– 26 ) + 5*x*.

##### Answer key

- One Solution i.e.
*x*= 27. - Infinite Solutions.
- Zero Solution.
- Infinite Solutions.
- One Solution i.e.
*x*= – 8. - Zero Solution.

+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. eTutorWorld provides the best K-12 Online Tutoring Services so you can learn from the comfort and safety of your home at an affordable cost.

Be it an exam, class test or a quiz, eTutorWorld’s Math, Science and English tutors are responsible for your academic progress. Meet your personal coach at your convenient day and time to get help for Grade 3-12 Math, Science and English subjects and AP, SAT, SSAT and SCAT Test Prep help and test practice. All our tutors are graduates and bring with them years of teaching experience to the tutoring lessons.

Our ‘*Learning by Design*’ methodology makes sure that each student is at the center of the teaching-learning process. All tutoring sessions start with a question to the student ‘*What do you want to learn today?*’ Hence, tutors diagnose your skills and recognize your requirements before the actual tutoring happens. Post every tutoring session, an individualized worksheet is emailed to the student to assimilate learned concepts. Regular formative assessments are used to evaluate a student’s understanding of the subject.

The state of the art technology used is stable, user friendly and safe. All you need is a computer or a tablet and an internet connection. The easy-to-use web conferencing software requires a one time download, using which the student can talk and chat to the tutor, annotate on an interactive shared whiteboard or even share documents, assignments or worksheets.

All tutoring sessions are recorded and made available for a month so you can review concepts taught.

Email or call our support team with any issues or questions – we are here for you 24X7.

Also, download free printable math and science worksheets in pdf format and solve SCAT and SSAT Practice Tests online. Sign up for a Free Trial Lesson Today!

Thousands have taken the eTutorWorld Advantage – what are you waiting for?

© 2019 eTutorWorld - Online Tutoring and Test Prep | All rights reserved