Solving Systems of Equations That Contain 3 Unknown Variables

The following is a link to the solution of Problem #4 in 7.3 Problems on Edmodo.
http://www.educreations.com/lesson/view/problem-4/15521447/?s=bt50Vf&ref=app

In this example, I will show the steps to solving a system of equations with 3 unknown variables:

4x + y - 3z = 11
2x - 3y +2z = 9
x + y + z = -3

First, we can start by eliminating y from the first two equations. To do this, we need to multiply the top equation by 3.

12x + 3y - 9z = 33
2x - 3y + 2z = 9

After adding these two equations together, we are left with the following:

14x - 7z = 42

Now, we can eliminate y from the two bottom equations. To do this, we need to multiply the bottom equation by 3.

2x - 3y + 2z = 9
3x + 3y + 3z = -9

After adding these two equations together, we are left with the following:

5x + 5z = 0

Next we will eliminate z by using the equation we produces earlier. We will need to multiply the first equation by 5 and the second by 7.

5 * (14x - 7z = 42) = 70x - 35z = 210
7 * (5x + 5z = 0) = 35x + 35z = 0

After adding these two equations together, we are left with the following:

105x = 210
x = 2

Now, we can substitute the value of x, which we have determined is 2, into either of the two equations we produced earlier. I will substitute x into the second one.

5x + 5z = 0
5(2) + 5z = 0
10 + 5z = 0
5z = -10
z = -2

Next, we can substitute the values of x and z into any of the original three equations. I will substitute x and z into the third one.

x + y + z = -3
(2) + y + (-2) = -3
y = -3

Finally, we have determined the values of all 3 variables. Before writing the answer down, we should confirm the answer by substituting these values into one of the first three equations. I will substitute them into the first equation.

4x + y - 3z = 11
4(2) + (-3) - 3(-2) = 11
8 - 3 + 6 = 11
11 = 11

Since the values we found for the variables were valid substitutions for this equation, we can conclude that we have found the correct values. The last step is to present our answer in the following format:
(x, y, z)

Therefore, our final answer is:
(2, -3, 2)