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# Solving Number Problems Involving System of Equations In Two Variables

Solving Number Problems Involving System of Equations In Two Variables

One of the most common applications of equations are number problems. This hub presents five number problems involving system of equations in two variables.

Problem Number One :

The sum of two numbers is 70. The smaller number increased by twenty equals the bigger number. What are these two numbers ?

Solution :

Let X = smaller number

Let Y = bigger number

Equation one : X + Y = 70

Equation Two: X + 20 = Y

Substitute equation 2 in equation one :

X + X + 20 = 70

2X + 20 = 70

2X = 70 – 20

2X = 50

(1/2) 2X = 50 (1/2)

X = 25

Y = 25 + 20 = 45

The two numbers are 25 and 45

Problem Number Two :

The sum of two numbers is 100. The bigger number is four times as large as the smaller number. Find the two numbers.

Solution :

Let X = smaller number

Let Y = bigger number

Equation one : X + Y = 100

Equation two: Y = 4X

Substituting Equation two in equation one :

X + 4X = 100

5X = 100

(1/5) 5X = 100 (1/5)

X = 20

Y = 4(20) = 80

The two numbers are 20 and 80

Problem Number Three :

A number is thrice as large as the other number. The smaller number is 25 less than half the larger number. Find these numbers.

Solution:

Let X = smaller number

Let Y = bigger number

Equation one : Y = 3X

Equation Two : X = y/2 – 25

Substituting equation one in equation two :

X = 3X/2 - 25

2 ( X = 3X/2 - 25) 2

2X = 3X - 50

2X – 3X = -50

-1 ( -X = - 50 )

X = 50

Y = 3 (50) = 150

The two numbers are 50 and 150

Sample Problem Number Four :

A certain number is ten times as large as the other number. This number is 50 greater than eight times the smaller number. Find these two numbers.

Solution :

Let X = smaller number

Let Y = the larger number

Equation one: Y = 10X

Equation Two : Y = 8X + 50

Substituting Equation one in equation two :

10X = 8X + 50

10X – 8X = 50

2X + 50

(1/2) 2X = 50 (1/2)

X = 25

Y = 10 (25) = 250

The two numbers are 25 and 250

Sample Problem Number Five :

The sum of two numbers is 70. The smaller number increased by 90 is equal to thrice the bigger number. Find these two numbers.

Solution :

Let X = smaller number

Let Y = bigger number

Equation one : X + Y = 70

Equation Two : X + 90 = 3Y

We will solve this using the elimination method by subtraction

X + Y = 70

(-) X - (+) 3Y =(+) -90

(1/4) 4Y = 160 (1/4)

Y = 40

X = 70 – 40 = 30

The two numbers we are looking for are 30 and 40.