SOLVING WORD PROBLEMS USING SYSTEMS OF EQUATIONS

Problem 1 :

Two numbers have a sum of 200 and a difference of 37. Find the numbers.

Solution :

Let x and y be two numbers.

x + y = 200 --- > (1)

x - y = 37 --- > (2)

From (2)

y = x - 37

Substitute y = x - 37 in (1)

x + (x - 37) = 200

x + x - 37 = 200

2x - 37 = 200

2x = 200 + 37

2x = 237

x = 237/2

x = 118 1/2

By applying x = 237/2 in (1)

237/2 + y = 200

y = 200 - 237/2

y = (400 - 237) / 2

y = 163/2

y = 81 1/2

So, two numbers are 118 1/2 and 81 1/2.

Problem 2 :

The difference between two numbers is 84 and their sum is 278. What is the numbers?

Solution :

x - y = 84 --- > (1)

x + y = 278 --- > (2)

From (1)

y = x - 84

Substitute y = x - 84 in (2)

x + (x - 84) = 278

x + x - 84 = 278

2x - 84 = 278

2x = 278 + 84

2x = 362

x = 362/2

x = 181

By applying x = 181 in (1)

181 - y = 84

-y = 84 - 181

-y = -97

y = 97

So, two numbers are 181 and 97. 

Problem 3:     

One number exceeds another by 11. The sum of the two numbers is 5. What are the numbers?

Solution :

x - y = 11 --- > (1)

x + y = 5 --- > (2)

From (1)

y = x - 11

Substitute y = x - 11 in (2)

x + (x - 11) = 5

x + x - 11 = 5

2x = 5 + 11

2x = 16

x = 8

By applying x = 8 in (2)

8 + y = 5

y = 5 - 8

y = -3

So, the numbers are 8 and -3.

Problem 4 :

The larger of two numbers is four times the smaller and their sum is 85. Find the two numbers.  

Solution :

Let x be the smaller number  and y be the larger number.

y = 4x --->(1)

x + y = 85 --->(2)

Substitute y = 4x in (2)

x + 4x = 85

5x = 85

x = 17

By applying x = 17 in (2)

17 + y = 85

y = 85 - 17

y = 68

So, two numbers are 17 and 68.