# How To Solve Linear Equation Questions Quickly

## How to solve Linear Equation Questions Quickly

We can define linear equation is the equation between two variables that gives a straight line when plotted on a graph  . A line or Linear Equation is defined as y = mx +c

### How to Solve Linear Equation Question & Definition

• A linear equation is an equation where variable quantities are in the first power only and whose graph is a straight line.
• The above line represent as,  y= mx + c

### Type 1: How To Solve Linear Equations Questions Quickly. Find the value of x or y

Question 1. If 3a + 7b = 75 and 5a – 5b = 25, what is the value of a + b?

Options:

A. 11

B. 6

C. 5

D. 17

Solution:    3a + 7b = 75 ……(1)

5a – 5b = 25 (divide the equation by 5)

we get, a – b = 5 …….(2)

Now multiplying eq. (2) by 7

and add to eq. (1), we get

3a + 7b = 75

7a – 7b = 35

On solving

10a = 110

a = $\frac{110}{10}$

a = 11

Now put the value of a in eq (2)

11 – b = 5

b = 11 – 5

b = 6

Therefore, a = 11 and b = 6

The value of a + b = 6 + 11 = 17

Correct option: D

Question 2. If 2x + y = 16 and 16x – y = 2, then find the value of x?

Options:

A. $\frac{1}{4}$

B. $\frac{17}{4}$

C. $\frac{17}{8}$

D. 4

Solution:     Given, 2x + y = 16

2x + y = 24

x + y = 4….(1)

Now, 16x – y = 2

(24) x – y = 2¹

x – y = $\frac{1}{4}$ ….(2)

On solving equation 1 and 2

We get,

2x = $\frac{17}{4}$

x = $\frac{17}{4 × 2}$ = $\frac{17}{8}$

Correct option: C

Question 3. The system of equations 3a + 5b = 6 and 6a + 10y = 6 has

Options:

A. No solution

B. One solution

C. Two solution

D. Infinite solution

Solution:     $\frac{a_{1}}{a_{2}}$ = 3/6 = 1/2

$\frac{b_{1}}{b_{2}}$= 5/10 = 1/2

$\frac{c_{1}}{c_{2}}$ =  $\frac{6}{6}$= 1

$\frac{a_{1}}{a_{2}}$ $\frac{b_{1}}{b_{2}}$  $\frac{c_{1}}{c_{2}}$

Therefore, there is no solution

Correct option: A

### Type 2: Solve Quickly Linear Equations Questions.Based on Word problems

Question 1. The difference between the two numbers is 45. The ratio of the two numbers is 8:3. Find the two numbers?

Options:

A. 72 and 27

B. 90 and 45

C. 81 and 36

D. 60 and 15

Solution:    Let the first number be 8x

Let the second number be 3x

Now, the difference between the two numbers is 45

Therefore, 8x – 3x = 45

5x = 45

x =  $\frac{45}{5}$

x = 9

Now, put the value of x in

8x = 8 × 9 = 72

3x = 3 × 9 = 27

Correct option: A

Question 2. The breadth of a rectangle is twice its length. If the perimeter of the rectangle is 84m. Then, calculate the length and breadth of the rectangle?

Options:

A. L=12 and B= 24

B. L = 14 and B = 28

C. L = 28 and B = 14

D. L = 24 and B = 12

Solution:    Perimeter of rectangle = 2 (l+b)

Length of the rectangle = x

Breadth of the rectangle = 2x

Perimeter of the rectangle = 84

2 (x + 2x) = 84

2 (3x) = 84

6x = 84

x =  $\frac{84}{6}$

x = 14

Therefore, the Length of the rectangle = 14m

And Breadth of the rectangle = 14 × 2 = 28m

Correct option: B

Question 3. Ajay bought 5 tickets for two concerts A and B and 10 tickets for concert A and C. He paid Rs. 350. Now the total of a ticket for concert A and B and ticket of A and C is Rs. 42, then what is the ticket price for concert A and B?

Options:

A. Rs. 10

B. Rs. 42

C. Rs. 14

D. Rs. 28

Solution:    Let the ticket price of concert A and B = a

Let the ticket price of concert A and C = b

According to the question, a + b = 42…… (1)

Ticket bought by Ajay = 5a + 10b = 350

= a + 2b = 70……(2)

Now solve equation 1 and 2

a + b = 42

a + 2b = 70

b = 70 – 42

b = 28

Now put the value of b in equation 1

a+ 28 = 42

a = 42 – 28

a = 14

Hence, the ticket price for concert A and B = Rs. 14

Correct option: C