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Goal: Solve systems of linear equations in three variables
Example 1 Use the Linear Combination Method
Solve the system.
SOLUTION
STEP 1 Rewrite the system as a system in two variables. First, add 2 times Equation 2 to Equation 1 to eliminate y.
Equation 1112y3x =+
Equation 24y2x =–
4z+
3z+
3y5x – 1=5z+ – Equation 3
= =
112y3x =+
4y2x –
4z+
3z+
112y3x =+
82y4x –
4z+
6z+
197x =10z+ New Equation 1
Example 1 Use the Linear Combination Method
Now add 3 times Equation 2 to Equation 3 to eliminate y.–
4y2x =– 3z+ 3y6x =+ 9z
13y5x =5z+– – 13y5x =5z+– –
– – 12–
New Equation 2x =4z– – 13–
STEP 2 Solve the new system of linear equations in two variables. First, add 7 times new Equation 2 to new Equation 1 to eliminate x.
= =
197x =10z+
x 4z– – 13–
197x =10z+
7x 28z– – 91–
=18z– 72–
Example 1 Use the Linear Combination Method
Solve for z.=z 4
Substitute 4 for z in new Equation 1 or 2 and solve for x to get x 3.–=
STEP 3 Substitute 3 for x and 4 for z in one of the original equations and solve for y.
–
4y2x =– 3z+ Equation 2
4y2 =– 3+ Substitute 3 for x and 4 for z.–( )4( )3–
4y =– 12+ Multiply.6–
4y =– 6+ Combine like terms.
2= Solve for y.y
Example 1 Use the Linear Combination Method
STEP 4 Check by substituting 3 for x, 2 for y, and 4 for z in each of the original equations.
–
ANSWER
The solution is x 3, y 2, and z 4, or the ordered triple ( 3, 2, 4).
–= = =–
f. (1,1,-3) g. (-2, 4, 0) h. (0, -3, 10) j. (-3, 0, -2)
Example 2 Solve a System with No Solution
Solve the system. Equation 12yx =+
Equation 2143y3x =+
z+
3z+
2yx – 4=z+ Equation 3
SOLUTION
Multiply Equation 1 by 3 and add the result to Equation 2.–
143y3x =+ 3z+ to Equation 2.
Add 3 times Equation 163y3x =3z– –– – –
8=0 False statement
Example 2 Solve a System with No Solution
ANSWER
Because solving the system resulted in the false statement 0 8, the original system of equations has no solution.
=
Example 3 Solve a System with Infinitely Many Solutions
Solve the system. Equation 14yx =+
Equation 2
z+
Equation 3
4yx =+ z–
123y3x =+ z+
SOLUTION
STEP 1 Rewrite the system as a system in two variables.
Add Equation 14yx =+ z+
to Equation 2.4yx =+ z–
New Equation 182y2x =+
Example 3 Solve a System with Infinitely Many Solutions
Add Equation 2
New Equation 2164y4x =+
4yx =+ z–
to Equation 3.123y3x =+ z+
STEP 2 Solve the new system of linear equations in two variables.
Add 2 times new Equation 1164y4x =–– – –
to new Equation 2.164y4x =+
00 =
Example 3 Solve a System with Infinitely Many Solutions
ANSWER
Because solving the system resulted in the true statement 0 0, the original system of equations has infinitely many solutions. The three planes intersect in a line.
=
Checkpoint
Tell how many solutions the linear system has. If the system has one solution, solve the system. Then check your solution.
Solve Systems
1yx =+ 3z+
1yx =+ z–
13yx =4z+–– –
1.
52y2x =2z+–
2. 4yx =z+–
2y3x =+ 2z– –
ANSWER 1; (1, 0, 0)
ANSWER no solution
x + y + 3z = 1x + y – z = 1-x – 3y + 4z = -1
x – y + z = 43x + y – 2z = -22x – 2y + 2z = 5
x + y + 2z = 10-x + 2y + z = 5-x + 4y + 3z = 15
Checkpoint
Tell how many solutions the linear system has. If the system has one solution, solve the system. Then check your solution.
Solve Systems
10yx =+ 2z+3.
ANSWER infinitely many solutions
52yx =z++–
154yx =3z++–
