There are 2 numbers. Twice the first number minus the second number equals 18. Twice the second number minus the first number equals 9. Find the numbers.

Question

There are 2 numbers. Twice the first number minus the second number equals 18. Twice the second number minus the first number equals 9. Find the numbers.

Answer 1

Let x = the first number

Let y = the second number

2x - y = 18     Multiply this equation by 2 and add to the second equation.

  2y - x = 9

-2y + 4x = 36

          3x = 45

            x = 15

 

2(15) - y = 18

30 - y = 18

12 = y

 

Answer 2

There are two numbers

 

ok, we have a and b.

 

Twice the first number (2*a) minus the second number (2*a - b) equals 18.

 

so 2*a - b = 18 (*)

 

Twice the second number (2*b) minus the first number (2*b - a) equals 9.

 

so 2*b - a = 9. OR -a + 2*b = 9 (**)

 

This is a system of linear equations. We can start by multiplying the second equation (**) by 2 which will give us.

 

2(-a + 2*b = 9)

 

-2*a + 4*b = 18

 

We did this so that when we add (*) to our new equation, we're left with just the variable b.

 

so

    (2*a -     b = 18)

+(-2*a + 4*b = 18) 

 

which leaves 3*b = 36

now divide both sides by 3 to get b = 12, now we just have to use our original equations with this value of b to find a.

 

2*a - b = 2*a - 12 = 18

 

add 12 to both sides

 

2*a = 30

 

divide both sides by 2

 

a = 15

 

So the answer is that a = 15 and b = 12.

 

We can check this answer by plugging these values into the original equations (*) and (**)

2*15 - 12 = 10 - 12 = 18

 

2*12 - 15 = 24 - 15 = 9.