Systems of 3 Equations and 3 Variables Lesson 29 Pages Lesson 29 Pages

Warm-up Problems A system of simultaneous equations that have no common solution is called ?. Evaluate the expression -2x – 5y when x = -5 and y = -3 Solve the system of equations using any convenient strategy: x + y = 5; x – y = 1

Warm-up Problems A system of simultaneous equations that have no common solution is called consistent. Evaluate the expression -2x – 5y when x = -5 and y = -3 -2(- 5) – 5(-3) = = 25 Solve the system of equations using any convenient strategy: x + y = 5; x – y = 1 Solution: (3, 2)

Activator #1 Three planes can intersect in a ? _, as shown below. Point

Activator #2 Three planes can intersect in a __, as shown below. Line

Activator #3 Describe the intersection of the 3 planes below. No intersection (parallel)

New Concepts Systems of equations are not limited to 2 equations in 2 variables. A linear equation in three variables is of the form Ax + By + Cz = D

New Concepts (cont’d) Systems of equations in 3 variables require 3 equations to solve. The solution, if it exists, to a solution of 3 equations in 3 variables is an ordered triple and is noted (x, y, z).

Steps to Solve a system of 3 equations in 3 variables 1. Use the two equations to eliminate one of the variables. 2. Use two other equations to eliminate the same variable as in Step 1. From Steps 1 & 2, there is now a system of two equations in two variables. 3. Solve the system of equations resulting from Steps 1 and 2.

Steps to Solve a system of 3 equations in 3 variables (cont’d) 4. Substitute the values for the two variables found in Step 3 into one of the original equations to solve for the third variable. 5. Check the solution by substituting the values for the three variables into each of the three original equations.

Classifying systems of 3 equations ‣ If a solution exists, the system of equations is called consistent & independent. ‣ If no solution exists (if the lines do not cross; they are parallel or skew), the system of equations is called inconsistent & independent. ‣ If infinitely many solutions exist (the lines are the same), the system of equations is called consistent & dependent.

Ex. 1a One Solution Justify that the solution to the system of equations is (1, 2, 5). x + y + z = 8 -x - 2y + z = 0 -2x + y + z = = (2) + 5 = 0 -2(1) = 5

Ex. 1 One Solution Solve the system of equations. x + y + z = 4 9x + 3y + z = 0 4x + 2y + z = 1 Solution: (1, -6, 9)

Ex. 2 Infinitely Many Solutions Justify that there are infinitely many solutions to the system of equations. 2x + 3y + 4z = 12 -6x – 12y – 8z = -56 4x + 6y + 8z = 24

Ex. 3 No Solutions Justify that there is no solution to the system of equations. x + 2y – 3z = 4 2x + 4y – 6z = 3 -x + 5y + 3z = 1

Ex. 4 Classify a System of Equations Solve the system of linear equations. Classify the system as Consistent or Inconsistent. -x – 5y + 3z = 7 5x + y + 3 z = 9 2x + 10y – 6z = -4

Using a Graphing Calculator Graphing calculators will simplify your work. x + 2y + z = 1 2x – y – 4 z = 1 x – y – z = 2 can be represented by: x y z 2 [A] [X] = [B] and can be solved by: [X] = [A] -1 [B] 2 1

Practice Solve the Lesson Practice Problems, p. 212 a-e Homework: p. 212 #1-23 odd.

Exit Ticket: p. 212 #2 Solve the system of equations. Classify the system as consistent or inconsistent; dependent or independent. -4x + 2y – z = 2 4x + 8y – 6z = 0 4x – 2y + z = 2