3.6 Systems with Three Variables Solving Three-Variable Systems by Substitution

2) Solving Three-Variable Systems by Substitution Substitution may be easier in cases where an equation can be solved for one variable Example: x + 32y – 32.7z = -382 Rewrite as: x = -382 – 32y + 32.7z

2) Solving Three-Variable Systems by Substitution Example 1: Solve the system by substitution. x – 2y + z = -4 -4x + y – 2z = 1 2x + 2y – z = 10 {

2) Solving Three-Variable Systems by Substitution Example 1: Solve the system by substitution. x – 2y + z = -4 -4x + y – 2z = 1 2x + 2y – z = 10 Re-write (1) as x =. { 1 2 3

2) Solving Three-Variable Systems by Substitution Example 1: Solve the system by substitution. x – 2y + z = -4 -4x + y – 2z = 1 2x + 2y – z = 10 Re-write (1) as x =. x = -4 + 2y – z { 1 2 3 1

2) Solving Three-Variable Systems by Substitution Example 1: Solve the system by substitution. x – 2y + z = -4 -4x + y – 2z = 1 2x + 2y – z = 10 Now…it gets ugly. Sub x = -4 + 2y – z in { 1 2 3 2

2) Solving Three-Variable Systems by Substitution Example 1: Solve the system by substitution. x – 2y + z = -4 -4x + y – 2z = 1 2x + 2y – z = 10 Sub x = -4 + 2y – z in { 1 2 3 2 2

2) Solving Three-Variable Systems by Substitution Example 1: Solve the system by substitution. x – 2y + z = -4 -4x + y – 2z = 1 2x + 2y – z = 10 Sub x = -4 + 2y – z in -4(-4 + 2y – z) + y – 2z = 1 { 1 2 3 2 2

2) Solving Three-Variable Systems by Substitution Example 1: Solve the system by substitution. x – 2y + z = -4 -4x + y – 2z = 1 2x + 2y – z = 10 Sub x = -4 + 2y – z in -4(-4 + 2y – z) + y – 2z = 1 16 – 8y + 4z + y – 2z = 1 -7y + 2z = -15 { 1 2 3 2 2 4

2) Solving Three-Variable Systems by Substitution Example 1: Solve the system by substitution. x – 2y + z = -4 -4x + y – 2z = 1 2x + 2y – z = 10 Sub x = -4 + 2y – z in { 1 2 3 3 3

2) Solving Three-Variable Systems by Substitution Example 1: Solve the system by substitution. x – 2y + z = -4 -4x + y – 2z = 1 2x + 2y – z = 10 Sub x = -4 + 2y – z 2(-4 + 2y – z) + 2y – z = 10 { 1 2 3 3 3

2) Solving Three-Variable Systems by Substitution Example 1: Solve the system by substitution. x – 2y + z = -4 -4x + y – 2z = 1 2x + 2y – z = 10 Sub x = -4 + 2y – z 2(-4 + 2y – z) + 2y – z = 10 -8 + 4y – 2z + 2y – z = 10 6y – 3z = 18 { 1 2 3 3 3 5

2) Solving Three-Variable Systems by Substitution Example 1: Solve the system by substitution. x – 2y + z = -4 -4x + y – 2z = 1 2x + 2y – z = 10 Solve for y and z with elimination, substitution. -7y + 2z = -15 6y – 3z = 18 { 1 2 3 4 5

2) Solving Three-Variable Systems by Substitution Example 1: Solve the system by substitution. x – 2y + z = -4 -4x + y – 2z = 1 2x + 2y – z = 10 Solve for y and z with elimination, substitution. -7y + 2z = -15 x 3 6y – 3z = 18 x 2 { 1 2 3 4 5

2) Solving Three-Variable Systems by Substitution Example 1: Solve the system by substitution. x – 2y + z = -4 -4x + y – 2z = 1 2x + 2y – z = 10 Solve for y and z with elimination, substitution. -7y + 2z = -15 x 3 -21y + 6z = -45 6y – 3z = 18 x 2 12y – 6z = 36 { 1 2 3 4 5

2) Solving Three-Variable Systems by Substitution Example 1: Solve the system by substitution. x – 2y + z = -4 -4x + y – 2z = 1 2x + 2y – z = 10 Solve for y and z with elimination, substitution. -7y + 2z = -15 x 3 -21y + 6z = -45 6y – 3z = 18 x 2 12y – 6z = 36 -9y = -9 y = 1 { 1 2 3 4 5

2) Solving Three-Variable Systems by Substitution Example 1: Solve the system by substitution. x – 2y + z = -4 -4x + y – 2z = 1 2x + 2y – z = 10 Solve for y and z with elimination, substitution. -7y + 2z = -15 -7(1) + 2z = -15 2z = -8 z = -4 { 1 2 3 4 4 Sub y = 1 in

2) Solving Three-Variable Systems by Substitution Example 1: Solve the system by substitution. x – 2y + z = -4 -4x + y – 2z = 1 2x + 2y – z = 10 Solve for x. x = -4 + 2y – z x = -4 + 2(1) –(-4) x = -4 + 2 + 4 x = 2 { 1 2 3 Sub y = 1, z = -4.

2) Solving Three-Variable Systems by Substitution Example 1: Solve the system by substitution. x – 2y + z = -4 -4x + y – 2z = 1 2x + 2y – z = 10 Therefore, the solution is (2, 1, -4). { 1 2 3

2) Solving Three-Variable Systems by Substitution Example 2: Solve the system by substitution. 12x + 7y + 5z = 16 -2x + y – 14z = -9 -3x - 2y + 9z = -12 {

2) Solving Three-Variable Systems by Substitution Example 2: Solve the system by substitution. 12x + 7y + 5z = 16 -2x + y – 14z = -9 -3x - 2y + 9z = -12 Rearrange . Sub in and . { 1 2 3 2 1 3

2) Solving Three-Variable Systems by Substitution Example 2: Solve the system by substitution. 12x + 7y + 5z = 16 -2x + y – 14z = -9 -3x - 2y + 9z = -12 Rearrange . Sub in and . y = -9 + 2x + 14z { 1 2 3 2 1 3 2

2) Solving Three-Variable Systems by Substitution Example 2: Solve the system by substitution. 12x + 7y + 5z = 16 -2x + y – 14z = -9 -3x - 2y + 9z = -12 Rearrange . Sub in and . y = -9 + 2x + 14z 12x + 7y + 5z = 16 { 1 2 3 2 1 3 2 1

2) Solving Three-Variable Systems by Substitution Example 2: Solve the system by substitution. 12x + 7y + 5z = 16 -2x + y – 14z = -9 -3x - 2y + 9z = -12 Rearrange . Sub in and . y = -9 + 2x + 14z 12x + 7y + 5z = 16 12x + 7(-9 + 2x + 14z) + 5z = 16 12x - 63 + 14x + 98z + 5z = 16 26x + 103z = 79 { 1 2 3 2 1 3 2 1 4

2) Solving Three-Variable Systems by Substitution Example 2: Solve the system by substitution. 12x + 7y + 5z = 16 -2x + y – 14z = -9 -3x - 2y + 9z = -12 Rearrange . Sub in and . y = -9 + 2x + 14z -3x - 2y + 9z = -12 -3x - 2(-9 + 2x + 14z) + 9z = -12 { 1 2 3 2 1 3 2 3

2) Solving Three-Variable Systems by Substitution Example 2: Solve the system by substitution. 12x + 7y + 5z = 16 -2x + y – 14z = -9 -3x - 2y + 9z = -12 Rearrange . Sub in and . y = -9 + 2x + 14z -3x - 2y + 9z = -12 -3x - 2(-9 + 2x + 14z) + 9z = -12 -3x + 18 – 4x – 28z + 9z = -12 -7x - 19z = -30 { 1 2 3 2 1 3 2 3 5

2) Solving Three-Variable Systems by Substitution Example 2: Solve the system by substitution. 12x + 7y + 5z = 16 -2x + y – 14z = -9 -3x - 2y + 9z = -12 Add and . 26x + 103z = 79 -7x - 19z = -30 { 1 2 3 4 5 4 5

2) Solving Three-Variable Systems by Substitution Example 2: Solve the system by substitution. 12x + 7y + 5z = 16 -2x + y – 14z = -9 -3x - 2y + 9z = -12 Add and . 26x + 103z = 79 x 7 182x + 721z = 553 -7x - 19z = -30 x 26 -182x – 494z = -780 { 1 2 3 4 5 4 5

2) Solving Three-Variable Systems by Substitution Example 2: Solve the system by substitution. 12x + 7y + 5z = 16 -2x + y – 14z = -9 -3x - 2y + 9z = -12 Add and . 26x + 103z = 79 x 7 182x + 721z = 553 -7x - 19z = -30 x 26 -182x – 494z = -780 227z = -227 z = -1 { 1 2 3 4 5 4 5

2) Solving Three-Variable Systems by Substitution Example 2: Solve the system by substitution. 12x + 7y + 5z = 16 -2x + y – 14z = -9 -3x - 2y + 9z = -12 Sub z = -1 in 26x + 103z = 79 26x + 103(-1) = 79 x = 7 { 1 2 3 4

2) Solving Three-Variable Systems by Substitution Example 2: Solve the system by substitution. 12x + 7y + 5z = 16 -2x + y – 14z = -9 -3x - 2y + 9z = -12 Sub z = -1 and x = 7 in 12x + 7y + 5z = 16 12(7) + 7y + 5(-1) = 16 84 + 7y – 5 = 16 y = -9 { 1 2 3 1

2) Solving Three-Variable Systems by Substitution Example 2: Solve the system by substitution. 12x + 7y + 5z = 16 -2x + y – 14z = -9 -3x - 2y + 9z = -12 Therefore, the solution is (7, -9, -1). { 1 2 3

Homework p.157 #10-12, 28, 29, 38, 39 ASSIGNMENT: 3.5, 3.6 on Monday Oct 19 QUIZ: 3.5, 3.6 on Wednesday Oct 21 TEST: 3.1, 3.2, 3.5, 3.6 on Monday Oct 26 Halloween Party: Friday October 30 5-8pm Dancing, food and games