here is the example problem
x + 2y – z=–3
2x – 2y + 2z = 8
2x – y + 3z = 9
Step 1: Identify or Create Opposite Coefficients
Identify or create opposite coefficients in two of the equations and add them vertically. Recall that opposite coefficients allow you to eliminate variables since they have a sum of zero. Do you see any opposite coefficients? Look at the first and second equations. There is a pair of opposite coefficients in 2y and –2y. Let’s add the first and second equations.
x + 2y – z=–3
+ 2x – 2y + 2z = 8
3x + 0y + z = 5 which simplifies to… 3x + z = 5
Unfortunately, you can’t find the value of one variable yet. Put this new equation off to the side for now, and move on to the next step.
Step 2: Identify or Create Opposite Coefficients AGAIN
This step is incredibly important! In this step, you must eliminate y again by combining two equations. But this time, you must use the equation you didn’t use in step 1.
In the first and third equations, the y terms have opposite signs. So these two equations are a good choice for elimination. Multiply each term of the third equation by 2.
x + 2y – z = –3 x + 2y – z = –3
2(2x – y + 3z) = 9 4x – 2y + 6z = 18
Now add the two equations.
x + 2y – z=–3
+ 4x – 2y + 6z = 18
5x + 0y + 5z = 15 which simplifies to…5x + 5z = 15
Again, you're left with an equation with two variables instead of one. But if you go back to the new equation from Step 1, you have two equations with the same two variables. You know how to take care of that!
Step 3: Solve the New System
A new system of equations, with only two variables, has been created by eliminating y in Steps 1 and 2.
3x + z=5
5x + 5z = 15
Now this looks familiar! You can solve this system of equations using the elimination or substitution method. The substitution method looks easier since z in the first equation has a coefficient of 1. Isolate the z variable in the first equation.
3x + z =5
–3x –3x
z = –3x + 5
Substitute –3x + 5 for z in the equation 5x + 5z = 15 and solve for x.
5x + 5z=15
5x + 5(–3x + 5) = 15
5x – 15x + 25 = 15
–10x + 25 = 15
–25 –25
–10x = –10
x = 1
Substitute the value of x into one of the equations and solve for z.
3x + z=5
3(1) + z = 5
3 + z = 5
–3 –3
z = 2
Now you know that x = 1 and z = 2.
Two variables down, one to go!
Step 4: Substitute and Solve
Substitute x = 1 and z = 2 into one of the original equations and solve for the remaining variable (y). Write the solution as an ordered triple. Solve for y when x = 1 and z = 2.
x + 2y – z=–3
(1) + 2y – (2) = –3
1 + 2y – 2 = –3
–1 + 2y = –3
+1 +1
2y = –2
y = –1
Since x = 1, y = –1 and z = 2, the solution is (1, –1, 2).
Graphically, this represents the only point where the three planes intersect.