find the numbers

Question

The sum of three numbers is 9.
The third number is half the sum of the first and second numbers.
The second number is 2 less than the sum of the first and third numbers.
Find the numbers.

Answer 1

Assign variables for the numbers we desire to find.

 

a = first number

b = second number

c = third number

 

Next, set up equations using the variables and info given.

 

a + b + c = 9           eq1

c = (a + b)/2            eq2

b = (a + c) - 2          eq3

 

We have a system of equations. 

 

 

Subtract  (a + c) on both sides of eq3 so that it looks like eq1.

 

-a + b - c = - 2

 

 

We will work with eq1 and eq3 for now.

 

a + b + c = 9   

-a + b - c = - 2

 

 

We can add the two equations to eliminated the a and c terms.

 

2b = 7

 b = 3.5

 

 

Next, substitute this value of b into all equations so that we only have two unknown variables.

 

a + 3.5 + c = 9

a + c = 5.5            eq1

 

 

c = (a + 3.5) / 2

2c = a + 3.5           eq2

 

 

3.5 = (a + c) - 2       eq3

 

 

Next, we can work with eq1 and eq2.

 

a + c = 5.5

-a + 2c = 3.5

 

Add the two equations to eliminate the a terms.

 

3c = 9

  c = 3

 

Now, we go back to the original eq1 to solve for a.  Substitute the values of b and c.

 

a = 9 - b - c

a = 9 - 3.5 - 3

a = 2.5

 

 

Our numbers are

 

a = 2.5

b = 3.5

c = 3

 

To ensure our numbers are correct in value, we check them into the original eq2 and eq3.

 

c = (a + b)/2

3 = (2.5 + 3.5)/2

3 = 6 / 2

3 = 3

 

eq2 checks out.

 

 

b = (a + c) - 2

3.5 = (2.5 + 3) - 2

3.5 = 5.5 - 2

3.5 = 3.5

 

eq3 checks out.