Section6.2 Systems of Equations in Three Variables Number of - PowerPoint PPT Presentation

Section6.2 Systems of Equations in Three Variables

Number of Solutions of a Linear System Like the two variable case, when you have systems of more variables: You can have exactly one solution

Number of Solutions of a Linear System Like the two variable case, when you have systems of more variables: You can have exactly one solution You can have no (zero) solutions

Number of Solutions of a Linear System Like the two variable case, when you have systems of more variables: You can have exactly one solution You can have no (zero) solutions You can have infinitely many solutions.

Substitution Let’s solve this system: x − y + z = 0 y + 2 z = − 2 x + y − z = 2 1. Pick one equation and solve for a variable in that equation. Equation 2: y + 2 z = − 2 → y = − 2 z − 2

Substitution Let’s solve this system: x − y + z = 0 y + 2 z = − 2 x + y − z = 2 1. Pick one equation and solve for a variable in that equation. Equation 2: y + 2 z = − 2 → y = − 2 z − 2 2. Plug that into both of the other equations. x − ( − 2 z − 2) + z = 0 x + ( − 2 z − 2) − z = 2 x + 2 z + 2 + z = 0 x − 2 z − 2 − z = 2 x + 3 z = − 2 x − 3 z = 4

Substitution (continued) 3. From the last step you should have two new equations with only two variables. Treat these like their own “mini” system of equations and solve for those variables. x + 3 z = − 2 x − 3 z = 4 Plug z back in to find x : Plug x into Equation 1: Solve for x in Equation 2: x = 3 z + 4 (3 z + 4) + 3 z = − 2 x − 3 z = 4 x = 3( − 1) + 4 6 z = − 6 x = 3 z + 4 x = − 3 + 4 z = − 1 x = 1

Substitution (continued) 4. Use the values you’ve found to plug back in and get the last variable. y = − 2 z − 2 y = − 2 ( − 1) − 2 y = 2 − 2 y = 0 The solution is (1,0,-1).

Elimination Let’s solve this system: x − y + 2 z = 2 3 x + y + 5 z = 8 2 x − y − 2 z = − 7 1. Create two pairs of equations and eliminate the same variable from both pairs. Equations 1 and 2: Equations 2 and 3: x − y +2 z = 2 3 x + y +5 z = 8 3 x + y +5 z = 8 2 x − 2 z = − 7 − y 4 x +7 z = 10 5 x +3 z = 1

Elimination (continued) 2. From the last step you should have two new equations with only two variables. Treat these like their own “mini” system of equations and solve for those variables. 4 x + 7 z = 10 5 x + 3 z = 1 Get x coefficients to Solve for x : match: Eliminate x : 5 x + 3 z = 1 5(4 x + 7 z ) = 5(10) 20 x +35 z = 50 5 x + 3(2) = 1 − 20 x − 12 z = − 4 20 x + 35 z = 50 5 x + 6 = 1 23 z = 46 5 x = − 5 − 4(5 x + 3 z ) = − 4(1) z = 2 x = − 1 − 20 x − 12 z = − 4

Elimination (continued) 3. Use the values you’ve found to plug back in and get the last variable. x − y + 2 z = 2 − 1 − y + 2(2) = 2 − y + 3 = 2 − y = − 1 y = 1 The solution is (-1,1,2).

Examples Solve the following systems of equations: 1. − x − y − 2 z = − 5 − x + 2 y + 7 z = 4 2 x + y + z = 7

Examples Solve the following systems of equations: 1. − x − y − 2 z = − 5 − x + 2 y + 7 z = 4 2 x + y + z = 7 (2 + z , 3 − 3 z , z )

Examples Solve the following systems of equations: 1. 2. − x − y − 2 z = − 5 w − y + z = 2 − x + 2 y + 7 z = 4 2 w − 2 x + y + z = 1 2 x + y + z = 7 − 2 x + y = 1 w + y = − 1 (2 + z , 3 − 3 z , z )

Examples Solve the following systems of equations: 1. 2. − x − y − 2 z = − 5 w − y + z = 2 − x + 2 y + 7 z = 4 2 w − 2 x + y + z = 1 2 x + y + z = 7 − 2 x + y = 1 w + y = − 1 (2 + z , 3 − 3 z , z ) no solution

Examples (continued) 3. 3 p + 2 r = 11 q − 7 r = 4 p − 6 q = 1

Examples (continued) 3. 3 p + 2 r = 11 q − 7 r = 4 p − 6 q = 1 � 4 , 1 2 , − 1 � 2

✩ ✩ ✩ Examples (continued) 4. Orange juice, an egg sandwich, and a cup of coffee from a local breakfast shop cost a total of ✩ 6.50. The owner posts a notice announcing that, effective the following week, the price of orange juice will increase 25%, and the price of egg sandwiches will increase 20%. After the increase, the same purchase will cost a total of ✩ 7.60, and orange juice will cost ✩ 1 more than coffee. Find the price of each item before the increase.

Examples (continued) 4. Orange juice, an egg sandwich, and a cup of coffee from a local breakfast shop cost a total of ✩ 6.50. The owner posts a notice announcing that, effective the following week, the price of orange juice will increase 25%, and the price of egg sandwiches will increase 20%. After the increase, the same purchase will cost a total of ✩ 7.60, and orange juice will cost ✩ 1 more than coffee. Find the price of each item before the increase. Orange juice: ✩ 2; egg sandwich: ✩ 3; coffee: ✩ 1.50