Systems of Linear Equations in Three Variables

Question

The sum of three numbers is 11. The sum of twice the first number, 3 times the second number, and 4 times the third number is 32. The difference between 3 times the first number and the second number is -2. Find the three numbers.

Answer 1

Define your variables: Let the three numbers be x, y, and z.

Just convert each phrase into a math sentence (equation):

x + y + z = 11

2x + 3y + 4z = 32

3x - y = -2

Using the last equation, solve for "y":

y = 3x + 2

Now substitute "3x + 2" into the first 2 equations wherever you see a "y":

x + (3x + 2) + z = 11 or 4x + z = 9

2x + 3(3x + 2) + 4z = 32 or 11x + 4z = 26

Using "4x + z = 9", solve for "z":

z = 9 - 4x

Plug "9 - 4x" in for "z' into the other equation:

11x + 4(9 - 4x) = 26

11x + 36 - 16x = 26

-5x = -10

x = 2

Using "z = 9 - 4x" plug in x = 2 to get z = 1

Using x + y + z = 11, plug in x = 2 and z = 1 to get y = 8

Answer 2