#### Linear Equations

Questions based on Linear equations are an important component of the CAT and XAT exam while the importance is low for MAT, CMAT, IRMA, and other management entrance exams hence your ability to formulate and solve equations is a key skill in the development of your thought process for Quantitative aptitude. As you go through with this topic, focus on understanding the core concepts, and create a framework in your mind that can help solve questions based on equations. For cracking CAT and other management exams Linear equation is a basic topic in Quantitative aptitude. This is the easiest of all and you can get a full score in these questions in one attempt.

#### CAT :

 YEAR NO. OF QUESTIONS GOOD ATTEMPT DIFFICULTY LEVEL 2019 2 2-3 Moderate 2018 4 3-4 Moderate 2017 3 2-3 Moderate

#### NMAT

 YEAR NO. OF QUESTIONS GOOD ATTEMPT DIFFICULTY LEVEL 2020 3 2-3 Easy 2019 3 2 Moderate 2018 4 2-3 Moderate

XAT

 YEAR NO. OF QUESTIONS GOOD ATTEMPT DIFFICULTY LEVEL 2019 1 1 Easy 2018 2 1 Moderate 2017 2 1-2 Moderate

IIFT

 YEAR NO. OF QUESTIONS GOOD ATTEMPT DIFFICULTY LEVEL 2019 2 2 Easy 2018 2 1 Moderate 2017 3 3 Easy

SNAP

 YEAR NO. OF QUESTIONS GOOD ATTEMPT DIFFICULTY LEVEL 2019 2 2 Easy 2018 1 1 Moderate 2017 2 1 Moderate

### Understanding Concepts

The best way to understand this topic is to practice hard and going through the concepts as mentioned below:

An equation whose solution is a straight line and the degree of the equation is equal to 1.

For Example

2 a + 4 = 0 — equation (1)

Solving a particular linear equation means finding a solution to an unknown variable that is usually on but can be several. In the above equation x is the unknown but there can be more than 2 or more variables (unknown) in a linear equation as given below.

6x + 2 y = 24 — (2)

6 x + 4y +10 z =12 — (3)

So, A linear equation is an equation that can be written in the form y= ax + b where x and y are variables & a and b are constants. Note that the exponential power raised to the variable of a linear equation is always 1.

These are examples of linear expressions:

x + 6

2 x + 3

2 x + 5 y

To solve linear equations, we need to care for the following concepts while solving it.

1. We can add a number, c, to both sides of the equation without changing the equation i.e. if a=b then by adding constant c we can write the equation as a+c = b+c
2. Similarly, we can subtract a number, c, from both sides of an equation i.e. if a=b then a-c = b-c
3.  Moreover, like addition and subtraction, we can also multiply both sides of an equation by a number, c, without changing the equation i.e. if a = b then it can be written as ac = bc for any value of c
4. both sides of an equation can be divided by a non-zero number, c, without changing the equation i.e. if a=b then a/c =b/c
These facts form the basis for solving all kinds of linear equations that we will be looking so it is very important that you know and do not forget these concepts used in linear equations.

### Previous year questions

Q1.) In an examination, Rama’s score was one-twelfth of the sum of the scores of Mohan and Anjali. After a review, the score of each of them increased by 6. The revised scores of Anjali, Mohan, and Rama were in the ratio 11:10:3. Then Anjali’s score exceeded Rama’s score by [CAT 2019 Slot 2]

1. 24
2. 26
3. 35
4. 32

Let score of Rama = r, Mohan = m and Anjali = a

As the score was one-twelfth of the sum of the scores of Mohan and Anjali. So 12r = m + a…… (1)

Similarly, after the score of each of them increased by 6, a score of scores of Anjali, Mohan, and Rama Becomes in the ratio 11: 10 :3 Let 11k = r+6, m+6 = 10k and a+6 = 3k, putting values of the 3 in equation 1,

we get a – r = 40 – 8 = 32

2. x + 4|y| = 33. How many integer values of (x, y) are possible? [CAT 2019 Slot 2]

1. 6
2. 3
3. 4
4. More than 6

First, we will reframe the equation:

3x = 33 – 4|y|

Since there are two integers x and y, and since |y| is always positive regardless of the sign of y, this means that when you subtract a multiple of 4 from 33, you need to get a multiple of 3.

Since 33 is already a multiple of 3, to obtain another multiple of 3, you will have to subtract a multiple of 3 from it. So, y has to be a positive or a negative multiple of 3.

y = 3, -3, 6, -6, 9, -9, 12, -12…etc.

For every value of y, x will have a corresponding integer value.

So there are infinite integer values possible for x and y.

Referring to the question back again we can conclude that the answer is “More than 6”

Choice D is the correct answer.

3. If the number of Old visitors buying Platinum tickets was equal to the number of Middle-aged visitors buying Economy tickets, then the number of Old visitors buying Gold tickets was _______? [CAT 2018 Slot 2]

Let Old – platinum = Middle aged – Economy = x We get x + 2a = 20 and a + x + 38 = 55 By solving these two equations we get x = 3. Ans: 3

4. Three friends, returning from a movie, stopped to eat at a restaurant. After dinner, they paid their bill and noticed a bowl of mints at the front counter. Sita took 1/3 of the mints but returned four because she had a momentary pang of guilt. Fatima then took 1/4 of what was left but returned three for similar reasons. Eshwari then took half of the remainder but threw two back into the bowl. The bowl had only 17 minutes left when the raid was over. How many mints were originally in the bowl? [CAT 2018 Slot 1]

a. 38
b. 31
c. 41
d. None of these (CAT 2001)

Let’s the initial count be X

Sita took 1/3 and returned four =  Current count is X – X/3 + 4 = 2X/3 + 4

Fatima took 1/4 and returned three = Current count is 3/4 * (2X/3 + 4) + 3 = X/2 + 3 + 3 = X/2 + 6

Eshwari took half of remaining and returned two =  1/2 (X/2 + 6) + 2 = X/4 + 3 + 2 = X/4 + 5

It is given that X/4 + 5 = 17

X/4 = 12

X = 48

### Preparation Phases in Linear Equations

Linear Equations is also an important part of Arithmetic which are generally framed in the Quants section of CAT, XAT, SNAP, NMAT, and other various management entrance exams We will divide the preparation of this topic into three phases following which can help the aspirants to score exceptionally well in the management entrance exams.

Phase I: In Phase, I the candidate can start with basic concept understanding and should try solving the basic questions from the topic Linear Equations. For this aspirant can follow Arun Sharma’s book and videos on Quantitative aptitude and after following the concept can solve the questions under Level of Difficulty 1 from the topic.

Phase II: In Phase II aspirants can start practicing questions from Level of difficulty 2 questions from Arun Sharma’s book from the given topic Linear Equation and also practice through the sectional test as time-based practice and analysis of the test is also necessary by this time.

Phase III: In Phase III aspirants can practice the topic from previous year papers of prominent management entrance exams like CAT, XAT, SNAP, NMAT, and various other exams. Aspirants can also practice through a mock test and analyse it thoroughly. Also, in this phase aspirants should go through the concepts once again at the same time should try solving advanced problems from the Linear equations.