Systems of Equations Back-Substitution: 3x3 Eliminating one variable Eliminating two variables...
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Transcript of Systems of Equations Back-Substitution: 3x3 Eliminating one variable Eliminating two variables...
Systems of Equations
Back-Substitution: 3x3
Eliminating one variable
Eliminating two variables
Copyright © 2011 Lynda Aguirre 1
Using Back-SubstitutionUsing Back-SubstitutionAn equation can only be solved when there is only one unknown variable
Back-substitution steps:1) Using an equation with only one variable, solve it for that variable
Copyright © 2011 Lynda Aguirre 2
2) Plug that value into another equation to find a second variable
3) Plug both values into the third equation to find the third variable.
Using Back-SubstitutionUsing Back-SubstitutionAn equation can only be solved when there is only one unknown variable
Copyright © 2011 Lynda Aguirre 3
Back-substitution steps:1) Using an equation with only one variable, solve it for that variable2) Plug that value into another equation to find a second variable3) Plug both values into the third equation to find the third variable.
Eliminating a VariableEliminating a Variable
Examine the situation:-This system does not have an equation with only one variable, but it has two equations with 2 variables.
We need a preparation step before we can use back-substitution
-Then use back-substitution in the original 3x3 system of equations
Process: -Using equations 2 and 3, use either substitution or elimination to wipe out one of the variables (y or z).
Copyright © 2011 Lynda Aguirre 4
Eliminating a VariableEliminating a Variable
Examine the situation:-This system does not have an equation with only one variable, but it has two equations with 2 variables.
We need a preparation step before we can use back-substitution
-Then use back-substitution in the original 3x3 system of equations
Process: -Using equations 2 and 3, use either substitution or elimination to wipe out one of the variables (y or z).
Copyright © 2011 Lynda Aguirre 5
Eliminating a VariableEliminating a Variable
Examine the situation:-This system does not have an equation with only one variable, but it has two equations with 2 variables.
We need a preparation step before we can use back-substitution
-Then use back-substitution in the original 3x3 system of equations
Process: -Using equations 2 and 3, use either substitution or elimination to wipe out one of the variables (y or z).
Copyright © 2011 Lynda Aguirre 6
Eliminating a VariableEliminating a Variable
Examine the situation:-This system does not have an equation with only one variable, but it has two equations with 2 variables.
We need a preparation step before we can use back-substitution
-Then use back-substitution in the original 3x3 system of equations
Process: -Using equations 2 and 3, use either substitution or elimination to wipe out one of the variables (y or z).
Copyright © 2011 Lynda Aguirre 7
Eliminating Two VariablesEliminating Two Variables
Examine the situation:- This system does not have an equation with only one variable, -It also does not have two equations with the same 2 variables.
We need several preparation steps before we can use back-substitution
We need to create 2 equations with 2 variables by wiping out the third variable.Step 1: First decide whether to wipe out the x’s, y’s or z’sStep 2: Using any two equations, use either substitution or elimination to wipe out the variable you chose.Step 3: Using the third equation and either of the two already used, use substitution or elimination to wipe out the same variable as in step 2.
Copyright © 2011 Lynda Aguirre 8
Eliminating Two VariablesEliminating Two Variables
Step 3: Using the third equation and either of the two already used, use substitution or elimination to wipe out the same variable (i.e. wipe out the x’s again)
Copyright © 2011 Lynda Aguirre 9
Step 1: First decide whether to wipe out the
x’s, y’s or z’sMy choice: Wipe out the x’s
Step 2: Using any two equations, use either substitution or elimination to wipe out the variable you chose.
My choice: Use equations 1 and 2 and
the elimination method to wipe out x’s.
We have created one equation with two variables ( y and z)
Use equations 1 and 3(since it hasn’t been used
yet) and the elimination method to wipe out x’s.
We have created another equation with the same two
variables ( y and z)
Now take these two new equations (in two variables) and eliminate another
variable
Now take these two new equations (in two variables) and eliminate another
variable
Copyright © 2011 Lynda Aguirre 10
Eliminating Two VariablesEliminating Two Variables
Step 4: Now use these two equations and either substitution or elimination to wipe out one of the remaining variables (y or z).
My choice: Use the elimination method to
wipe out the z’s.
Now do back-substitution into either one of the equations above
Copyright © 2011 Lynda Aguirre 11
Eliminating Two VariablesEliminating Two Variables
My choice: Plug y=1 into the top equation
to find z.
Now do back-substitution into any of the original three equations (with y=1 and z=2)
ORIGINAL 3 EQUATIONS
My choice: Plug y=1 and z=2 into the top equation to find x.