Solving systems with three equations and three unknowns

1
Section 7.2

• Solving systems with three equations and three
  unknowns

2
Substitution Method

• 1) Look to see if any of the equations are
  missing any variables.
• 2) If one is, then solve for one of the
  remaining two variables in terms of the other
  one.
• 3) Use substitution to eliminate that variable
  in the other two equations.
• 4) Solve the resulting system of two equations
  with two variables the way you normally solve
  one.
• 5) Substitute the values of those two variables
  into one of the original equations.

3
Example

• 2x 3y 4z -1
• 5x y 13
• x y z 14

2x 3y 4z -1
The second equation has no z, so solve that
equation for y.
5x y 13
x y z 14
5x y 13
Save this equation and do the same thing with the
3rd equation
5x 13 y
Plug this into the first equation
x y z 14
2x 3(5x 13) 4z -1
x 5x 13 z 14
2x 15x 39 4z -1
13x 39 4z -1
6x 13 z 14
13x 4z -40
13x 4z -40
6x z 27
6x z 27
Now solve this system
4

• 6x z 27-13x 4z -40

Solve the first equation for z
6x z 27
z -6x 27
z -6x 27
-13x 4(-6x 27) -40
Plug this into the second equation
-13x -24x 108 -40
-37x 108 -40
z -6(4) 27
-37x -40 108
z -24 27
-37x -148
z 3
-37 -37
x 4
Plug this back into the equation above
5
Now, remember that original system? Plug these
two values into one of the equations and solve
for the remaining variable.

• 2x 3y 4z -1
• 5x y 13
• x y z 14

x y z 14
4 y 3 14
y 7 14
y 7
solution (4, 7, 3)
6
If none of the equations are missing a variable,
then you have to do it the long way.

• 1. Choose a variable that would be easy or
  convenient to solve for (has a coefficient of 1
  or -1) in one of the equations.
• 2. Solve that equation for that variable.
• 3. Substitute that expression in to the other
  two equations in place of that variable.
• 4. Solve the resulting system of 2 equations
  with 2 unknowns.
• 5. Plug in those two values into one of the
  original equations to solve for the third
  variable.