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Solving Linear Equations Easily
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. Find Everything related to Linear Equations and How to Solve Linear Equation Quickly on this page.


Type : 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
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