Pair of Linear Equations – Notes, MCQs, Quiz & Worksheet

Let the smaller angle be x degrees.
And the larger angle will be (x + 18) degrees (since it exceeds the smaller by 18 degrees).

Two angles are supplementary if their sum is 180°. So, we have the equation:

x + (x + 18) = 180
2x + 18 = 180
x + 9 = 90 
x = 81

The larger angle is x + 18:
Larger angle = 81 + 18 = 99

Hence the smaller angle is 81°  &  the larger angle is 99°.

Let the cost of 1 pencil be Rs. x and the cost of 1 pen be Rs. y

From the Question:
5 pencils and 7 pens cost Rs. 50
=>  5x + 7y = 50   -------------------------------------------- (1)

7 pencils and 5 pens cost Rs. 46
=>  7x + 5y = 46    -------------------------------------------- (2)

Step 1:Eliminate one variable
Multiply both equations to make the coefficients of x the same:

Multiply the first equation by 7 then:
35x + 49y = 350 ------------------------------------- (3)

Multiply the second equation by 5:
=>  35x + 25y = 230  ----------------------------------(4)

Now equation(3) - eyuation(4)

=>   (35x + 49y) − (35x + 25y) = 350 − 230
=>    24y = 120
=>     y = 5

Step 2: Substitute y = 5 into equation (1)

 5x + 7y = 50
=>   5x + 7(5) = 50
=>   5x + 35 = 50
=>   5x = 15
=>    x = 3

Hence cost of one pencil = Rs. 3 & Cost of one pen = Rs. 5 

Let the two-digit number have:

Tens digit = x
Units digit = y
So the number is: 10x + y

It is given that:
The sum of the digits is 9:
So   x + y = 9   ----------------------------------(1)

If 27 is added, the digits reverse:
10x + y + 27 = 10y + x 
9x - 9y = -27
x - y = -3    ---------------------------------------(2)

Now solve the two equations:

Equation 1:   x + y = 9
Equation 2:    x - y = -3

Add both equations:
(x + y) + (x - y) = 9 + (-3)
2x = 6 
=> x = 3
Substitute x = 3 into Equation 1:
3 + y = 9 
=>  y = 6

So, the number is: 10x + y = 10×3 + 6 = 36
Hence answer is 36

Let fixed charge for the first 3 days = Rs. x
And extra charge for each additional day = Rs. y

From the Question:
Saritha kept the book for 7 days, so she paid:
Fixed charge for first 3 days = x
Extra charge for remaining 4 days = 4y
So:  x + 4y = 27     ----------------------------------------- (1)

She kept the book for 5 days, so she paid:
Fixed charge for first 3 days = x
Extra charge for remaining 2 days = 2y
So:  x + 2y = 21    --------------------------------------------(2)

Now solve the equations:
From Equation 1:
x + 4y = 27

From Equation 2:
x + 2y = 21

Now subtract Equation 2 from Equation 1:
(x + 4y) − (x + 2y) = 27 − 21
=>  2y = 6
=>   y = 3

Now substitute y = 3 into Equation 2:
x + 2(3) = 21
x + 6 = 21
x = 15
Hence Fixed charge = Rs. 15   &   Extra charge per day = Rs. 3

Let the amount of money A has be Rs. x and B has Rs. y.

First condition:  If A gives Rs. 30 to B:
A will have x - 30
B will have y + 30

According to the condition:
y + 30 = 2(x - 30)     ------------------------------------------------ (1)

Second condition:  If B gives Rs. 10 to A:
A will have x + 10
B will have y - 10

According to the condition:
x + 10 = 3(y - 10)   ---------------------------------------------------- (2)

Now solve the equations:

From Equation (1):
y + 30 = 2x - 60
y = 2x - 90    -----------------------------------------------(3)

Substitute y value  into Equation (2):
x + 10 = 3((2x - 90) - 10)
x + 10 = 3(2x - 100)
x + 10 = 6x - 300
310 = 5x
x = 62

Now to find y put x = 62 in equation (3):

y = 2(62) - 90 = 124 - 90 = 34

Hence A has Rs. 62 & B has Rs. 34

Let the first person has Rs. x
And the second person has Rs. y

The first person says, “Give me a hundred, friend! I shall then become twice as rich as you.”
This means: After the second person gives Rs. 100 to the first person, the first person will have x + 100, and the second person will have y - 100. According to the condition:
x + 100 = 2(y - 100)
x - 2y = -300  ------------------------------------ (1)

The second person says, “If you give me ten, I shall be six times as rich as you.”
This means: After the first person gives Rs. 10 to the second person, the first person will have x - 10, and the second person will have y + 10. According to the condition:
y + 10 = 6(x - 10)
- 6x + y = -70  ----------------------------------------(2)

Solve the two equations:
Multiply Equation 1 by 6 :

6x - 12y = -1800 ---------------------------------------- (3)

Now add Equation (2) and Equation (3):
-11y = -1870
=>  y = 170

Now substitute y = 170 into Equation 1:
x - 2(170) = -300
=> x - 340 = -300
=> x = 40

Hence the first person has Rs. 40  &  the second person has Rs. 194

Let the number of Rs. 50 notes be x
Let the number of Rs. 100 notes be y

From the Question:
The total amount is Rs. 2000
So: 50x + 100y = 2000
Divide the whole equation by 50:
=>   x + 2y = 40   -------------------------------------------- (1)

The total number of notes is 25
So: x + y = 25

Now solve the two equations:
From x + y = 25, we get:
x = 25 - y    ------------------------------------------------------- (2)
Substitute this into the first equation:
(25 - y) + 2y = 40
=>  25 + y = 40
=>   y = 15

Now put y = 15 into equation (2)
=>  x = 25 - 15 = 10

Hence Rs. 50 notes = 10   &   Rs. 100 notes = 15 

The ratio of both person incomes is 9 : 7
So, let their incomes be 9x and 7x

The ratio of their expenditures is 4 : 3
So, let their expenditures be 4y and 3y

Both save Rs. 2000 per month

Use the formula:
Income = Expenditure + Savings

For the first person:
Income = 9x, Expenditure = 4y, Saving = 2000
=>  9x = 4y + 2000   --------------------------------------------------- (1)

For the second person:
Income = 7x, Expenditure = 3y, Saving = 2000
=>  7x = 3y + 2000   ------------------------------------------------- (2)

Solve the two equations:
From the first equation:
=> 4y = 9x − 2000   ---------------------------------------- (3)

From the second equation:
=> 3y = 7x − 2000  ------------------------------------------ (4)

Now multiply:

Third equation by 3:
12y = 27x − 6000   ------------------------------------- (5)

Fourth equation by 4:
12y = 28x − 8000  ------------------------------------(6)

Now from (5)th and (6)th equations
=> 27x − 6000 = 28x − 8000
=>  2000 = x

Now find the incomes:
First person's income = 9x = 9 × 2000 = Rs. 18,000

Second person's income = 7x = 7 × 2000 = Rs. 14,000

Hence  First person's income = Rs. 18,000 & Second person's income = Rs. 14,000

Let the number of students in each row be x and the number of rows be y.
So, the total number of students is x × y.

First condition:
If 4 students are added to each row, the number of rows becomes 2 less.

This gives the equation:
(x + 4)(y - 2) = x × y
 =>   xy - 2x + 4y - 8 = xy
=>    -2x + 4y = 8    ------------------------------------ (1)

Second condition:
If 4 students are removed from each row, the number of rows becomes 4 more.

This gives the equation:
(x - 4)(y + 4) = x × y
=>   xy + 4x - 4y - 16 = xy
=>   4x - 4y = 16     ---------------------------------- (2)

Now add both equations:

(−2x + 4x) + (4y − 4y) = 8 + 16
=>  2x = 24  =>  x = 12

Now put x = 12 into Equation 1:

−2(12) + 4y = 8
=>   −24 + 4y = 8
=>   4y = 32   =>    y = 8

Hence  Students in each row = 12 & Number of rows = 8

Total students = 12 × 8 = 96

So, there are 96 students in the class.

Let the tens digit be x and the units digit be y.
Then the two-digit number is: 10x + y

First condition:
The number is 4 more than 6 times the sum of its digits.
So   10x + y = 6(x + y) + 4
=> 10x + y = 6x + 6y + 4
=> 10x - 6x + y - 6y = 4
=> 4x - 5y = 4   ---------------------------------(1)

Second condition:
If 18 is subtracted from the number, the digits are reversed.
The reversed number is: 10y + x
So   (10x + y) - 18 = 10y + x
=>  10x + y - 18 = 10y + x
=>  10x - x + y - 10y = 18
=>   9x - 9y = 18
=>    x - y = 2 ----------------------------------(2)

Solve the two equations:
From Equation 2:
x = y + 2
Substitute into Equation 1:
4(y + 2) - 5y = 4
4y + 8 - 5y = 4
-y + 8 = 4
y = 4

Then,   x = y + 2 = 4 + 2 = 6

So, the number is:
10x + y = 10×6 + 4 = 64

Let the numerator be x.
Then the denominator is x + 11 (since it is 11 more than the numerator).

So, the fraction is:  x / (x + 11)

According to the problem:
If 8 is added to both the numerator and the denominator, the fraction becomes 1/2.

That gives the equation:
(x + 8) / (x + 19) = 1/2

Now cross-multiply:
2(x + 8) = x + 19
2x + 16 = x + 19
x + 16 = 19
x = 3

So, the numerator is 3, and the denominator is:
x + 11 = 3 + 11 = 14

Final Answer: The fraction is 3/14.

Let the first number be x and the second number be y.

According to the question:

Three times the first number plus the second number is 142:
So, 3x + y = 142        -----------------------------(1)

Four times the first number exceeds the second number by 138:
So, 4x - y = 138       --------------------------------(2)

Now, add Equation (1) and Equation (2):

(3x + y) + (4x - y) = 142 + 138
3x + y + 4x - y = 280
7x = 280
x = 280 divided by 7
x = 40

Now substitute x = 40 into Equation (1) Then

3x + y = 142
3(40) + y = 142
120 + y = 142
y = 142 - 120
y = 22

Hence The first number is 40 & The second number is 22

Let the two numbers be x and y.

So we have two equations:
x + y = 137  ---------------------------(1)
x - y = 43   ------------------------------(2)
Add both equations:
(x + y) + (x - y) = 137 + 43
2x = 180
x = 90

Now substitute x = 90 into the first equation:
90 + y = 137
y = 137 - 90
y = 47
Hence the two numbers are 90 and 47.

It is given:
x + y = 5xy   ----------------------------- (1)
3x + 2y = 13xy    -----------------------------(2)
Also, x ≠ 0 and y ≠ 0

Step 1:   Divide both equations by xy
From equation 1:
x + y = 5xy
Divide both sides by xy:
(x/xy) + (y/xy) = 5
=>  1/y + 1/x = 5 ------------------------------------(3)

From equation 2:
3x + 2y = 13xy
Divide both sides by xy:
(3x/xy) + (2y/xy) = 13
=>  3/y + 2/x = 13 --------------------------------(4)

Step 2:Let
1/x = a and 1/y = b

Then Equation (3) becomes:
a + b = 5 ------------------------------------(5)
Equation (4) becomes:
2a + 3b = 13 -------------------------------------(6)

Step 3:Solve equations (5) and (6)

From Equation (3):   a = 5 - b
Substitute into Equation (6):
2(5 - b) + 3b = 13
10 - 2b + 3b = 13
10 + b = 13
b = 3 
now a = 5 - b = 5 - 3 = 2

Step 4: Back-substitute

a = 1/x = 2  =>  x = 1/2
b = 1/y = 3  =>  y = 1/3

Final Answer: x = 1/2 and y = 1/3

Let the two numbers be x and y,  And we assume where x is greater than y.

So according to the question:

The difference between the two numbers is 14:
So, x - y = 14     --------------------------------------------(1)

The difference between their squares is 448:
So, x² - y² = 448     -----------------------------------------(2)

As we know that x² - y² = (x - y)(x + y).
So put this into Equation (2):

(x - y)(x + y) = 448

From Equation (1),    x - y = 14.
Now substitute:

14(x + y) = 448

Divide both sides by 14:
x + y = 32     ----------------------------------------(3)

Now add equation (1) & (3)

(x - y) + (x + y) = 14 + 32
2x = 46
x = 23

Now substitute x = 23 into Equation 1:
23 - y = 14
y = 9

Hence the first number is 23  &   the second number is 9

Let fare from the bus stand to Malleswaram = Rs. x
And fare from the bus stand to Yeshwanthpur = Rs. y

From the Question:
2 tickets to Malleswaram and 3 to Yeshwanthpur cost Rs. 46
=>  2x + 3y = 46    ------------------------------------- (1)

3 tickets to Malleswaram and 5 to Yeshwanthpur cost Rs. 74
=>   3x + 5y = 74   -----------------------------------------(2)

Now solve the equations:
Multiply Equation 1 by 3:
=>  6x + 9y = 138   ----------------------------------(3)

Multiply Equation 2 by 2:
=> 6x + 10y = 148  -----------------------------------------------(4)

Now subtract Equation 3 from Equation 4:
(6x + 10y) − (6x + 9y) = 148 − 138
=>  y = 10
Now substitute y = 10 into Equation 1:
2x + 3(10) = 46
=> 2x + 30 = 46
=> 2x = 16
=> x = 8

Hence fare to Malleswaram = Rs. 8  &   fare to Yeshwanthpur = Rs. 10

We are given two equations:

x + y = 14   --------------------------------- (1)
x - y = 4    ------------------------------------(2)

Step 1:Add the two equations to eliminate y:
(x + y) + (x - y) = 14 + 4
=> 2x = 18
=> x = 9

Step 2: Substitute x = 9 into the first equation:
9 + y = 14
=>  y = 5
Hence, the solution is:   x = 9 and y = 5