Honors Algebra II 3.5 Solving Systems with Three Variables.

Solving Systems of Equations in 3 variables by Elimination: Example 1: Solve the system by elimination. (Number the equations to make it easier!) GOAL: Eliminate one of the variables at a time.

Solving Systems of Equations in 3 variables by Elimination: Example 1: Solve the system by elimination. (Number the equations to make it easier!) GOAL: Eliminate one of the variables at a time. Equation #1 Equation #2 Equation #3

Solving Systems of Equations in 3 variables by Elimination: Example 1: Solve the system by elimination. (Number the equations to make it easier!) GOAL: Eliminate one of the variables at a time. Equation #1 Equation #2 Equation #3 Equation #1 and #2 have the opposite coefficients for y, so that term can be easily eliminated.

Solving Systems of Equations in 3 variables by Elimination: Example 1: Solve the system by elimination. (Number the equations to make it easier!) GOAL: Eliminate one of the variables at a time. Equation #1 Equation #2 Equation #3 Equation #1 and #2 have the opposite coefficients for y, so that term can be easily eliminated. ZAP THE y-TERMS!!!

Solving Systems of Equations in 3 variables by Elimination: Example 1: Solve the system by elimination. (Number the equations to make it easier!) GOAL: Eliminate one of the variables at a time. Equation #1 Equation #2 Equation #3 Equation #1 and #2 have the opposite coefficients for y, so that term can be easily eliminated. ZAP THE y-TERMS!!! Equation #1 + Equation #2

Solving Systems of Equations in 3 variables by Elimination: Example 1: Solve the system by elimination. (Number the equations to make it easier!) GOAL: Eliminate one of the variables at a time. Equation #1 Equation #2 Equation #3 Equation #1 and #2 have the opposite coefficients for y, so that term can be easily eliminated. ZAP THE y-TERMS!!! Equation #1 + Equation #2

Solving Systems of Equations in 3 variables by Elimination: Example 1: Solve the system by elimination. (Number the equations to make it easier!) GOAL: Eliminate one of the variables at a time. Equation #1 Equation #2 Equation #3 Equation #2 and #3 have the opposite coefficients for y, so that term can be easily eliminated. Equation #1 + Equation #2 Equation #2 + Equation #3

Solving Systems of Equations in 3 variables by Elimination: Example 1: Solve the system by elimination. (Number the equations to make it easier!) GOAL: Eliminate one of the variables at a time. Equation #1 Equation #2 Equation #3 Equation #2 and #3 have the opposite coefficients for y, so that term can be easily eliminated. ZAP THE y-TERMS!!! Equation #1 + Equation #2 Equation #2 + Equation #3

Solving Systems of Equations in 3 variables by Elimination: Example 1: Solve the system by elimination. (Number the equations to make it easier!) GOAL: Eliminate one of the variables at a time. Equation #1 Equation #2 Equation #3 Equation #2 and #3 have the opposite coefficients for y, so that term can be easily eliminated. Equation #1 + Equation #2 Equation #2 + Equation #3

Solving Systems of Equations in 3 variables by Elimination: Example 1: Solve the system by elimination. (Number the equations to make it easier!) GOAL: Eliminate one of the variables at a time. Equation #1 Equation #2 Equation #3 Equation #2 and #3 have the opposite coefficients for y, so that term can be easily eliminated. Equation #1 + Equation #2 Equation #2 + Equation #3

Solving Systems of Equations in 3 variables by Elimination: Example 1: Solve the system by elimination. (Number the equations to make it easier!) GOAL: Eliminate one of the variables at a time. Equation #1 Equation #2 Equation #3 Equation #2 and #3 have the opposite coefficients for y, so that term can be easily eliminated. Equation #1 + Equation #2 Equation #2 + Equation #3 Now we have two equations with two variables… just like we’ve been doing!

Solving Systems of Equations in 3 variables by Elimination: Example 1: Solve the system by elimination. (Number the equations to make it easier!) GOAL: Eliminate one of the variables at a time. Equation #1 Equation #2 Equation #3 Equation #1 + #2 Equation #2 + #3

Solving Systems of Equations in 3 variables by Elimination: Example 1: Solve the system by elimination. (Number the equations to make it easier!) GOAL: Eliminate one of the variables at a time. Equation #1 Equation #2 Equation #3 Equation #1 + #2 Equation #2 + #3 The z-values are opposites, so add straight down to eliminate the z-terms. ZAP THE z-TERMS!!!

Solving Systems of Equations in 3 variables by Elimination: Example 1: Solve the system by elimination. (Number the equations to make it easier!) GOAL: Eliminate one of the variables at a time. Equation #1 Equation #2 Equation #3 Equation #1 + #2 Equation #2 + #3 The z-values are opposites, so add straight down to eliminate the z-terms. ZAP THE z-TERMS!!!

Solving Systems of Equations in 3 variables by Elimination: Example 1: Solve the system by elimination. (Number the equations to make it easier!) GOAL: Eliminate one of the variables at a time. Equation #1 Equation #2 Equation #3 Equation #1 + #2 Equation #2 + #3 The z-values are opposites, so add straight down to eliminate the z-terms. ZAP THE z-TERMS!!!

We now have one of the three values. Just two more to go!!! Solving Systems of Equations in 3 variables by Elimination: Example 1: Solve the system by elimination. (Number the equations to make it easier!) GOAL: Eliminate one of the variables at a time. Equation #1 Equation #2 Equation #3 Equation #1 + #2 Equation #2 + #3 The z-values are opposites, so add straight down to eliminate the z-terms. ZAP THE z-TERMS!!! We now have one of the three values. Just two more to go!!!

Solving Systems of Equations in 3 variables by Elimination: Example 1: Solve the system by elimination. (Number the equations to make it easier!) GOAL: Eliminate one of the variables at a time. Equation #1 Equation #2 Equation #3 Equation #1 + #2 Equation #2 + #3

Solving Systems of Equations in 3 variables by Elimination: Example 1: Solve the system by elimination. (Number the equations to make it easier!) GOAL: Eliminate one of the variables at a time. Equation #1 Equation #2 Equation #3 Equation #1 + #2 Equation #2 + #3 Plug the value for x back into one of the equations and solve for z.

Solving Systems of Equations in 3 variables by Elimination: Example 1: Solve the system by elimination. (Number the equations to make it easier!) GOAL: Eliminate one of the variables at a time. Equation #1 Equation #2 Equation #3 Equation #1 + #2 Equation #2 + #3 Plug the value for x back into one of the equations and solve for z.

Solving Systems of Equations in 3 variables by Elimination: Example 1: Solve the system by elimination. (Number the equations to make it easier!) GOAL: Eliminate one of the variables at a time. Equation #1 Equation #2 Equation #3 Equation #1 + #2 Equation #2 + #3 Plug the value for x back into one of the equations and solve for z.

We now have TWO of the three values. Just ONE more to go!!! Solving Systems of Equations in 3 variables by Elimination: Example 1: Solve the system by elimination. (Number the equations to make it easier!) GOAL: Eliminate one of the variables at a time. Equation #1 Equation #2 Equation #3 Equation #1 + #2 Equation #2 + #3 Plug the value for x back into one of the equations and solve for z. We now have TWO of the three values. Just ONE more to go!!!

Solving Systems of Equations in 3 variables by Elimination: Example 1: Solve the system by elimination. (Number the equations to make it easier!) GOAL: Eliminate one of the variables at a time. Equation #1 Equation #2 Equation #3 Plug both values back into one of the original equations and solve for the final value, y.

Solving Systems of Equations in 3 variables by Elimination: Example 1: Solve the system by elimination. (Number the equations to make it easier!) GOAL: Eliminate one of the variables at a time. Equation #1 Equation #2 Equation #3 Plug both values back into one of the original equations and solve for the final value, y.

Solving Systems of Equations in 3 variables by Elimination: Example 1: Solve the system by elimination. (Number the equations to make it easier!) GOAL: Eliminate one of the variables at a time. Equation #1 Equation #2 Equation #3 Plug both values back into one of the original equations and solve for the final value, y.

Solving Systems of Equations in 3 variables by Elimination: Example 1: Solve the system by elimination. (Number the equations to make it easier!) GOAL: Eliminate one of the variables at a time. Equation #1 Equation #2 Equation #3 Plug both values back into one of the original equations and solve for the final value, y.

Solving Systems of Equations in 3 variables by Elimination: Example 1: Solve the system by elimination. (Number the equations to make it easier!) GOAL: Eliminate one of the variables at a time. Equation #1 Equation #2 Equation #3 Plug both values back into one of the original equations and solve for the final value, y.

Solving Systems of Equations in 3 variables by Elimination: Example 1: Solve the system by elimination. (Number the equations to make it easier!) GOAL: Eliminate one of the variables at a time. Equation #1 Equation #2 Equation #3 Plug both values back into one of the original equations and solve for the final value, y.

We’ve found all three values!!! Solving Systems of Equations in 3 variables by Elimination: Example 1: Solve the system by elimination. (Number the equations to make it easier!) GOAL: Eliminate one of the variables at a time. Equation #1 Equation #2 Equation #3 WOO HOO!!! We’re done! We’ve found all three values!!! Our answer is (5, 1, -2)

Solving Systems of Equations in 3 variables by Elimination: Example 1: Solve the system by elimination. (Number the equations to make it easier!) GOAL: Eliminate one of the variables at a time.

Solving Systems of Equations in 3 variables by Elimination: Example 1: Solve the system by elimination. (Number the equations to make it easier!) GOAL: Eliminate one of the variables at a time. Equation #1 Equation #2 Equation #3

Solving Systems of Equations in 3 variables by Elimination: Example 1: Solve the system by elimination. (Number the equations to make it easier!) GOAL: Eliminate one of the variables at a time. ZAP THE y-TERMS!!! Equation #1 Equation #2 Equation #3

Solving Systems of Equations in 3 variables by Elimination: Example 1: Solve the system by elimination. (Number the equations to make it easier!) GOAL: Eliminate one of the variables at a time. ZAP THE y-TERMS!!! Equation #1 Equation #2 Equation #3 Equation #1 +Equation #2

Solving Systems of Equations in 3 variables by Elimination: Example 1: Solve the system by elimination. (Number the equations to make it easier!) GOAL: Eliminate one of the variables at a time. ZAP THE y-TERMS!!! Equation #1 Equation #2 Equation #3 Equation #1 +Equation #2

Solving Systems of Equations in 3 variables by Elimination: Example 1: Solve the system by elimination. (Number the equations to make it easier!) GOAL: Eliminate one of the variables at a time. ZAP THE y-TERMS!!! Equation #1 Equation #2 Equation #3 Equation #1 +Equation #2 Equation #1 +Equation #3

Solving Systems of Equations in 3 variables by Elimination: Example 1: Solve the system by elimination. (Number the equations to make it easier!) GOAL: Eliminate one of the variables at a time. ZAP THE y-TERMS!!! Equation #1 Equation #2 Equation #3 Equation #1 +Equation #2 Equation #1 +Equation #3

Solving Systems of Equations in 3 variables by Elimination: Example 1: Solve the system by elimination. (Number the equations to make it easier!) GOAL: Eliminate one of the variables at a time. ZAP THE y-TERMS!!! Equation #1 Equation #2 Equation #3 Equation #1 +Equation #2 Equation #1 +Equation #3

Solving Systems of Equations in 3 variables by Elimination: Example 1: Solve the system by elimination. (Number the equations to make it easier!) GOAL: Eliminate one of the variables at a time. Equation #1 Equation #2 Equation #3 Equation #1 + #2 Equation #1 + #3

Solving Systems of Equations in 3 variables by Elimination: Example 1: Solve the system by elimination. (Number the equations to make it easier!) GOAL: Eliminate one of the variables at a time. Equation #1 Equation #2 Equation #3 ZAP THE z-TERMS!!! Equation #1 + #2 Equation #1 + #3

Solving Systems of Equations in 3 variables by Elimination: Example 1: Solve the system by elimination. (Number the equations to make it easier!) GOAL: Eliminate one of the variables at a time. Equation #1 Equation #2 Equation #3 ZAP THE z-TERMS!!! Equation #1 + #2 Equation #1 + #3

Solving Systems of Equations in 3 variables by Elimination: Example 1: Solve the system by elimination. (Number the equations to make it easier!) GOAL: Eliminate one of the variables at a time. Equation #1 Equation #2 Equation #3 Equation #1 + #2 Equation #1 + #3 One down, two to go! We have our first value, x = 1.

Solving Systems of Equations in 3 variables by Elimination: Example 1: Solve the system by elimination. (Number the equations to make it easier!) GOAL: Eliminate one of the variables at a time. Equation #1 Equation #2 Equation #3 Equation #1 + #2 Equation #1 + #3

Solving Systems of Equations in 3 variables by Elimination: Example 1: Solve the system by elimination. (Number the equations to make it easier!) GOAL: Eliminate one of the variables at a time. Equation #1 Equation #2 Equation #3 Equation #1 + #2 Equation #1 + #3

Solving Systems of Equations in 3 variables by Elimination: Example 1: Solve the system by elimination. (Number the equations to make it easier!) GOAL: Eliminate one of the variables at a time. Equation #1 Equation #2 Equation #3 Equation #1 + #2 Equation #1 + #3

Solving Systems of Equations in 3 variables by Elimination: Example 1: Solve the system by elimination. (Number the equations to make it easier!) GOAL: Eliminate one of the variables at a time. Equation #1 Equation #2 Equation #3 Two down! We’re almost there!

Solving Systems of Equations in 3 variables by Elimination: Example 1: Solve the system by elimination. (Number the equations to make it easier!) GOAL: Eliminate one of the variables at a time. Equation #1 Equation #2 Equation #3

Solving Systems of Equations in 3 variables by Elimination: Example 1: Solve the system by elimination. (Number the equations to make it easier!) GOAL: Eliminate one of the variables at a time. Equation #1 Equation #2 Equation #3

Solving Systems of Equations in 3 variables by Elimination: Example 1: Solve the system by elimination. (Number the equations to make it easier!) GOAL: Eliminate one of the variables at a time. Equation #1 Equation #2 Equation #3

Solving Systems of Equations in 3 variables by Elimination: Example 1: Solve the system by elimination. (Number the equations to make it easier!) GOAL: Eliminate one of the variables at a time. Equation #1 Equation #2 Equation #3 Woo Hoo! We’re Done! (1, 2, -1)

Solving Systems of Equations in 3 variables by Elimination: Example 1: Solve the system by elimination. (Number the equations to make it easier!) GOAL: Eliminate one of the variables at a time.

Solving Systems of Equations in 3 variables by Elimination: Example 1: Solve the system by elimination. (Number the equations to make it easier!) GOAL: Eliminate one of the variables at a time. The solution is (-2, 6, -3)

Solving Systems of Equations in 3 variables by Elimination: Example 1: Solve the system by elimination. (Number the equations to make it easier!) GOAL: Eliminate one of the variables at a time.

Solving Systems of Equations in 3 variables by Elimination: Example 1: Solve the system by elimination. (Number the equations to make it easier!) GOAL: Eliminate one of the variables at a time. The solution is (2, -1, 1)

Solving Systems of Equations in 3 variables by Substitution: GOAL: Substitute for one of the variables to create a system with 2 variables!!!

Solving Systems of Equations in 3 variables by Substitution: Example 3: Solve the system by substitution. (Number the equations to make it easier!) GOAL: Substitute for one of the variables to create a system with 2 variables!!! The solution is (2, 1, -4)

Solving Systems of Equations in 3 variables by Substitution: Example 3: Solve the system by substitution. (Number the equations to make it easier!) GOAL: Substitute for one of the variables to create a system with 2 variables!!!

Solving Systems of Equations in 3 variables by Substitution: Example 3: Solve the system by substitution. (Number the equations to make it easier!) GOAL: Substitute for one of the variables to create a system with 2 variables!!! The solution is (4, 1, 5)