Transcript Three Equations Three Variables

SOLVING SYSTEMS OF THREE EQUATIONS WITH THREE VARIABLES
In this lesson you will study an algebraic method for solving
linear system of three equations and three variables. This
method is often referred to as the Elimination Method.
THE ELIMINATION METHOD
1
Choose two equations and combine them to
eliminate a variable.
2
Select different combination and combine to
eliminate the same variable.
3
Solve the resulting system of two equations and
two variables.
4
Substitute the values into any original equation to
find the value of the third variable.
x  2 y  z  1
2x  y  2z  4
2 x  y  z  4
Equation 1
Equation 2
Equation 3
Choose a variable and eliminate it by combining two equations.
2x  y  2z  4
Equation 2
2 x  y  z  4 Equation 3
4x
 z 0
Equation 4
Save Equation 4 for Step 3: Solving two
equations and two variables.
x  2 y  z  1
2x  y  2z  4
2 x  y  z  4
Equation 1
Equation 2
Equation 3
Select a DIFFERENT combination that will eliminate the SAME variable.
Equation 1
Twice
Equation 2
x  2 y  z  1
 x  2 y  z  1
2  2x  y  2z  4  4x  2 y  4z  8
Save Equation 5 for Step 3:
Solving two equations and
two variables.
5x
Equation 5

3z  7
Solve the new system for the remaining variables using
either linear combination or the substitution method.
3(Equation 4)
Equation 5
3  4 x  z  0  
5 x  3z  7

12 x  3 z  0
5 x  3z  7
7 x  7
x  1
Substitute the value of x into Equation 4 or 5 and solve for z
4  1  z  0
4  z  0
z4
Substitute the values of x and z
into any of the original equations
to find the value of y
Equation 3
x  2 y  z  1
2x  y  2z  4
2 x  y  z  4
Equation 1
Equation 2
Equation 3
2 x  y  z  4
2  1  y  4  4
2  y  4  4
The solution is
 1, 2, 4
y  6  4
y2
Your Turn. . .with a hint.
Try the “three box” method.
3 Variables/3 Eq.
2q  3r  2 s  11
q  2r  3s  11
3q  r  s  2
2 Variables/2 Eq.
1 Variables/1 Eq.
The Solution is . . .
q  2
r 3
s 1
Solve this . . .
2 x  5 y  z  23
x  2y  z  8
3x  y  2 z  17
x  2
y 3
z  4