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# 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

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 2 ^{x + y} = 16 and 16^{x – 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, 2^{x + y} = 16

2^{x + y} = 2^{4}

x + y = 4….(1)

Now, 16^{x – y} = 2

(2^{4})^{ 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

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**

**Read Also** – **Formulas to solve linear equation question**

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