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Systems of Equations Back-Substitution: 3x3 Eliminating one variable Eliminating two variables Copyright © 2011 Lynda Aguirre1

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Using Back-Substitution An 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 Aguirre2 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.

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Using Back-Substitution An equation can only be solved when there is only one unknown variable Copyright © 2011 Lynda Aguirre3 Back-substitution steps: 1) Using an equation with only one variable, solve it for that variable 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.

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Eliminating 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 Aguirre4

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Eliminating 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 Aguirre5

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Eliminating 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 Aguirre6

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Eliminating 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 Aguirre7

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Eliminating 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’s Step 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 Aguirre8

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Eliminating 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 Aguirre9 Step 1: First decide whether to wipe out the x’s, y’s or z’s My 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

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Copyright © 2011 Lynda Aguirre10 Eliminating 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

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Copyright © 2011 Lynda Aguirre11 Eliminating 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.

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Solving Systems of Equations: Elimination Method.

Solving Systems of Equations: Elimination Method.

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