Solving Systems of Linear Equations in Three Variables

1. Solving Systems of Linear Equations in Three Variables AII, 2.0: Students solve systems of linear equations and inequalities (in two or three variables) by substitution, with graphs, or with matrices. LA, 6.0: Students demonstrate an understanding that linear systems are inconsistent (have no solutions), have exactly one solution, or have infinitely many solutions

2. Before we begin Quiz on solving a system of two equations (30 Minutes)

3. Solving Systems of Linear Equations in Three Variables Objectives Key Words Ordered triple Linear combination System of equations • Solve systems of linear equations in three variables • Use the linear combination method • Solve a system with NO solution • Solve a system with Infinitely many solutions • Solve systems of three variables by substitution

4. The Linear Combination Method (3-Variable System) • REWRITE the linear system in three variables as a linear system in two variables using the linear combination method. • Combine two equations and eliminate one variable • Combine a different pair of equations to eliminate the same variable • SOLVE the new linear system for both of its variables. • Combine the equations that resulted from steps 1 and 2 to eliminate one of the two variables that are left • Solve for the remaining variable • Substitute that value into one of the equations with two variables • Solve for the second variable • SUBSTITUTE the values found in Step 2 into on of the original equations and solve for the remaining variable. • CHECK the solution in each of the original equations WRITE THIS DOWN!!! The steps to solving a system of linear equations in three variables. Example:

5. Example 1 SOLUTION STEP 1 Rewrite the system as a system in two variables. First, add 2 times Equation 2 to Equation 1 to eliminate y. 3x + 2y + 4z 11 3x + 2y + 4z 11 = = – – 3z 6z 2x y 4 4x 2y 8 + + = = New Equation1 7x 19 + 10z = Use the Linear Combination Method Solve the system. Equation 1 3x + 2y + 4z 11 = – Equation 2 2x y 4 + 3z = – – 5x 3y 1 Equation 3 + 5z =

6. Example 1 – – – – 5x 5x 3y 3y + + 5z 5z 1 1 = = – – + – – – – 3z 9z 2x y 4 6x 3y 12 13 13 + = = – – New Equation2 4z x = STEP 2 Solve the new system of linear equations in two variables. First, add 7 times new Equation 2 to new Equation 1 to eliminate x. 10z 10z 7x 19 7x 19 + + = = – – – – – 4z 28z x 7x 91 = = – – 72 18z = Use the Linear Combination Method Now add 3 times Equation 2 to Equation 3 to eliminate y. –

7. Example 1 Substitute 4 for z in new Equation 1 or 2 and solve for x to get x3. – = STEP 3 Substitute3 for x and 4 for z in one of the original equations and solve for y. – Equation2 – 3z 2x y 4 + = Substitute 3 for x and 4 for z. – 3 2 y 4 + – = – Multiply. – 12 6 y 4 + = ( ) ( ) – 3 4 Combine like terms. – y 4 + 6 = y Solve for y. 2 = Use the Linear Combination Method Solve forz. 4 z =

8. Example 1 ANSWER The solution is x3, y2, and z4, or the ordered triple ( 3, 2, 4). – = = = – Use the Linear Combination Method STEP 4 Check by substituting 3 for x, 2 for y, and 4 for z in each of the original equations. –

9. Example 2 SOLUTION Multiply Equation 1 by 3 and add the result to Equation 2. – – – – – Add 3 times Equation 1 3x 3y 6 3z = – to Equation 2. + + 3z 3x 3y 14 = False statement 0 8 = Solve a System with No Solution Solve the system. + Equation 1 x y + z 2 = Equation 2 3x + 3y 14 + 3z = – x 2y 4 Equation 3 + z =

10. Example 2 Solve a System with No Solution ANSWER Because solving the system resulted in the false statement 0 8, the original system of equations has no solution. =

11. Example 3 SOLUTION STEP 1 Rewrite the system as a system in two variables. Add Equation 1 x + y + z 4 = – to Equation 2. x + y z 4 = New Equation 1 2x + 2y 8 = Solve a System with Infinitely Many Solutions Solve the system. Equation 1 x + y + z 4 = – Equation 2 x + y z 4 = Equation 3 3x + 3y 12 + z =

12. Example 3 to Equation 3. New Equation 2 4x + 4y 16 = STEP 2 Solve the new system of linear equations in two variables. 3x + 3y 12 + z = – – – 4x 4y 16 Add 2 times new Equation 1 = – to new Equation 2. 4x + 4y 16 = 0 0 = Solve a System with Infinitely Many Solutions – Add Equation 2 x + y z 4 =

13. Example 3 Solve a System with Infinitely Many Solutions ANSWER Because solving the system resulted in the true statement 0 0, the original system of equations has infinitely many solutions. The three planes intersect in a line. =

14. Checkpoint x + y + 3z 1 = ANSWER – 1; (1, 0, 0) x + y z 1 = – 2. x y + z 4 = ANSWER – – no solution 3x + y 2z 2 = – 2x 2y 5 + 2z = Solve Systems Tell how many solutions the linear system has. If the system has one solution, solve the system. Then check your solution. 1. – – – x 3y 1 + 4z =

15. Checkpoint x + y + 2z 10 = – x 2y 5 + z + = ANSWER infinitely many solutions Solve Systems Tell how many solutions the linear system has. If the system has one solution, solve the system. Then check your solution. 3. – x 4y 15 + 3z + =

16. The Substitution Method (3-Variable System) • SOLVE for one of the variables in any of the equations • SUBSTITUTE the variable found in Step 1 into another equation. • SUBSTITUTE the variable found in Step 1 into the last equation. • SOLVE the new linear system of two equations in two variables for both of its variables. NOTE: use preferred method. • SUBSTITUTE the values found in Step 4 into original equation to find the last variable. • CHECK the solution in each of the original equations WRITE THIS DOWN!!! The steps to solving a system of linear equations in three variables. Example:

17. Conclusion Summary Assignment To be completed in class Pg156 #(14, 18, 22, 26, 31, 32) Extra Practice Assignment to be completed by the end of the week. Systems of two equations • How do you solve a system of linear equations in three variables? • Rewrite the system as a linear system in two variables by eliminating one variable. Solve that system for each variable. Substitute those two values into an equation in the original system to get the values of the third variable.