Describe the System with 3 Variables ?
This is an example of solving a system of equations using the substitution method. Warning: You will not understand this example if you try to read it quickly. Take your time, and try to understand where each step comes from.
Let's try a system of 3 equations, with 3 variables:
2x - y + z = -1 (3a)
x + z = -2 (3b)
-2x + y + z = -5 (3c)
Step 1: Solve the second equations for x. (We chose the second equation because it's simplest.)
x = -2 - z (4)
Step 2: Eliminate x from the other two equations, by substitution:
2x - y + z = -1
-2x + y + z = -5
2(-2 - z) - y + z = -1
-2(-2 - z) + y + z = -5
-4 -2z - y + z = -1
4 + 2z + y + z = -5
-y - z = 3 (5a)
y + 3z = -9 (5b)
Look! We've got it down to two variables instead of three!
Let's repeat steps 1 and 2, to get it down to one variable.
First, solve equation (5a) for y:
-y - z = 3
y = -3 - z (6)
Next, use substitution to eliminate y from (5b):
y + 3z = -9
(-3 - z) + 3z = -9
2z = -6
z = -3.
We've found the value of z ! We'll use this in equation (6) to find y:
y= -3 - z
= -3 - (-3)
= 0.
So we know z = -3 and y = 0. We'll use this in equation (4) to find x.
x= -2 - z (4)
= -2 -(-3)
=1.
Thus, the solution is x = 1, y = 0, z = -3.