3.6 Systems with Three Variables 1.Solving Three-Variable Systems by Elimination 2.Solving Three-Variable Systems by Substitution

1) Solving Three-Variable Systems by Elimination A three-variable system produces a 3D graph that is a plane A three equation system produces three planes The planes may never intersect, intersect once, or have an infinite number of intersections Use elimination to solve

1) Solving Three-Variable Systems by Elimination Example 1: Solve by elimination.x – 3y + 3z = -4 2x + 3y – z = 15 4x – 3y – z = {

1) Solving Three-Variable Systems by Elimination Example 1: Solve by elimination.x – 3y + 3z = -4 2x + 3y – z = 15 4x – 3y – z = You have to work with what you are given to eliminate one variable at a time. {

1) Solving Three-Variable Systems by Elimination Example 1: Solve by elimination.x – 3y + 3z = -4 2x + 3y – z = 15 4x – 3y – z = 19 Step 1:Add and to cancel y {

1) Solving Three-Variable Systems by Elimination Example 1: Solve by elimination.x – 3y + 3z = -4 2x + 3y – z = 15 4x – 3y – z = 19 Step 1:Add and to cancel y. x – 3y + 3z = -4 2x + 3y – z = {

1) Solving Three-Variable Systems by Elimination Example 1: Solve by elimination.x – 3y + 3z = -4 2x + 3y – z = 15 4x – 3y – z = 19 Step 1:Add and to cancel y. x – 3y + 3z = -4 2x + 3y – z = 15 3x + 2z = New two-variable equation {

1) Solving Three-Variable Systems by Elimination Example 1: Solve by elimination.x – 3y + 3z = -4 2x + 3y – z = 15 4x – 3y – z = 19 Step 1:Add and to cancel y {

1) Solving Three-Variable Systems by Elimination Example 1: Solve by elimination.x – 3y + 3z = -4 2x + 3y – z = 15 4x – 3y – z = 19 Step 1:Add and to cancel y. 2x + 3y – z = 15 4x – 3y – z = {

1) Solving Three-Variable Systems by Elimination Example 1: Solve by elimination.x – 3y + 3z = -4 2x + 3y – z = 15 4x – 3y – z = 19 Step 1:Add and to cancel y. 2x + 3y – z = 15 4x – 3y – z = 19 6x - 2z = New two-variable equation 3 {

1) Solving Three-Variable Systems by Elimination Example 1: Solve by elimination.x – 3y + 3z = -4 2x + 3y – z = 15 4x – 3y – z = 19 Step 2: Add and to solve for x {

1) Solving Three-Variable Systems by Elimination Example 1: Solve by elimination.x – 3y + 3z = -4 2x + 3y – z = 15 4x – 3y – z = 19 Step 2:Add and to solve for x. 3x + 2z = 11 6x - 2z = {

1) Solving Three-Variable Systems by Elimination Example 1: Solve by elimination.x – 3y + 3z = -4 2x + 3y – z = 15 4x – 3y – z = 19 Step 2:Add and to solve for x. 3x + 2z = 11 6x - 2z = 34 9x = 45 x = {

1) Solving Three-Variable Systems by Elimination Example 1: Solve by elimination.x – 3y + 3z = -4 2x + 3y – z = 15 4x – 3y – z = 19 Step 3: Sub x in to solve for z. 3(5) + 2z = 11 2z = -4 z = Sub x = 5 into 44 {

1) Solving Three-Variable Systems by Elimination Example 1: Solve by elimination.x – 3y + 3z = -4 2x + 3y – z = 15 4x – 3y – z = 19 Step 4:Sub x and z into one of the original equations. Solve for y. x – 3y + 3z = -4 5 – 3y + 3(-2) = -4 5 – 3y – 6 = -4 -3y = – 5 -3y = -3 y = Sub x = 5, z = -2 in 1 {

1) Solving Three-Variable Systems by Elimination Example 1: Solve by elimination.x – 3y + 3z = -4 2x + 3y – z = 15 4x – 3y – z = 19 Therefore, the solution is (5, 1, -2). 123 {

1) Solving Three-Variable Systems by Elimination Example 2: Solve by elimination.2x – y + z = 4 x + 3y – z = 11 4x + y – z = {

1) Solving Three-Variable Systems by Elimination Example 2: Solve by elimination.2x – y + z = 4 x + 3y – z = 11 4x + y – z = 14 Step 1:Add and to cancel z. 2x – y + z = 4 x + 3y – z = 11 3x + 2y = {

1) Solving Three-Variable Systems by Elimination Example 2: Solve by elimination.2x – y + z = 4 x + 3y – z = 11 4x + y – z = 14 Step 1:Subtract and to cancel z. x + 3y – z = 11 4x + y – z = 14 -3x + 2y = {

1) Solving Three-Variable Systems by Elimination Example 2: Solve by elimination.2x – y + z = 4 x + 3y – z = 11 4x + y – z = 14 Step 2:Use equations and to find x and y. 3x + 2y = 15 -3x + 2y = -3 4y = 12 y= {

1) Solving Three-Variable Systems by Elimination Example 2: Solve by elimination.2x – y + z = 4 x + 3y – z = 11 4x + y – z = 14 Step 2:Use equations and to find x and y. 3x + 2y = 15 3x + 2(3) = 15 3x + 6 = 15 3x = 9 x = Sub y = 3 in 44 {

1) Solving Three-Variable Systems by Elimination Example 2: Solve by elimination.2x – y + z = 4 x + 3y – z = 11 4x + y – z = 14 Step 3:Solve for the remaining unknown. 2x – y + z = 4 2(3) – 3 + z = 4 6 – 3 + z = 4 z = – 6 z = Sub x = 3 and y = 3 in 11 {

1) Solving Three-Variable Systems by Elimination Example 2: Solve by elimination.2x – y + z = 4 x + 3y – z = 11 4x + y – z = 14 Therefore, the solution is (3, 3, 1). 123 {

Example 3: Solve by elimination.-x + 2z = -9 -x – 3y – 4z = 2 -3x – 2y + 2z = 17 1) Solving Three-Variable Systems by Elimination {

Example 3: Solve by elimination.-x + 2z = -9 -x – 3y – 4z = 2 -3x – 2y + 2z = 17 Therefore, no unique solution. 1) Solving Three-Variable Systems by Elimination {

Summary of Steps 1)Elimination twice, create equations (4) and (5) 2)Solve for unknown a 3)Substitute a into equation (4) or (5) to find b 4)Substitute a and b into (1), (2) or (3) to find c

Homework p.157 #1-3, 25, 26