3.5 Solving Linear Systems in Three Variables. 10/4/13. Intersection of 3 planes. We’ve been solving system of equations in 2 variables. The solution is a point where the lines intersect. For systems of equations with 3 variables, the solution is a point where all 3 planes intersect. Solve:.
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3.5 Solving Linear Systems in Three Variables
10/4/13
We’ve been solving system of equations in 2 variables. The solution is a point where the lines intersect.
For systems of equations with 3 variables, the solution is a point where all 3 planes intersect.
Example 1
Equation 1
Equation 2
Equation 3
Notice Eqn1 has only 2 variables. Solve for one variable (x).
Substitute 3z +5 for x in the other 2 equations.
Dist. Prop
Combine Like terms
New Eqn 2
New Eqn 3
7( )
2( )
New Eqn 2
New Eqn 3
Solve by Elimination
Substitute z = 3 in
+
+
Solution (x, y, z)
(4, 1, 3)
Substitute z= 3 in any of the new Eqns.
Check the Solution (4, 1, 3)
Solve the system:
Step 1: Pick any 2 original equations and eliminate a variable. Eliminate the same variable from a second pair of original equations.
Step 2: With the 2 new equations from Step 1 eliminate one of the 2 variables and solve for the remaining variable. Substitute the value you obtained for the variable into one of the 2 new equations and solve for the other variable.
Step 3: Substitute the values of the 2 variables obtained in Step 2 into one of the 3 original equations and solve for the last variable (the one you eliminated in step 1).
Step 4: Check the solution in each of the original equations.
Example 2
Solve.
Step 1
New Eqn 1
Step 1
Step 2
New Eqn 2
Step 3
1( )
Example 3
Solve the system.
ANSWER
Equation 1
3x
+
2y
+
4z
11
=
( 3, 2, 4).
–
Equation 2
2x
y
4
+
3z
=
–
–
5x
3y
1
Equation 3
+
5z
=
x

y

z
3
=
ANSWER
+
(2, 2, 1)
x
+
2y
1
5z
=
Example 4
Solve the system. Then check your solution.
+
x
y
+
4z
=
4
Homework:
3.5 p.156 #7, 1619