Additional Practice – back side of worksheet #1 x = $ in 6% y = $ in 12% Objective Function: P =.06x +.12y Constraints:

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Additional Practice – back side of worksheet #1 x = $ in 6% y = $ in 12% Objective Function: P =.06x +.12y Constraints:

A storeowner wants to limit the weekly payroll to $960. Employees working regular hours receive $4 per hour, and employees working overtime receive $6 per hour. On the average the store makes $18 for each regular hour of employee work and $32 on each over time hour of employee work. If overtime hours are restricted to at most 2/3 of the regular hours, how should the owner schedule working hours to maximize profit? Objective Function: P = 18x + 32y Constraints: Additional Practice – back side of worksheet #2

Objective Function: P = 18x + 32y Constraints:

Solve Systems of Linear Equations in 3 Variables 1.7 (M3)

General Steps for Solving Systems with 3 variables 1.Combine 2 equations to make a new equation with 2 unknowns (eliminate 1 of the variables) 2.Do the same with 2 different equations (make sure you eliminate the same variable) 3.Solve the system of the 2 new equations from steps #1 and #2 4.Solve for 1 variable. 5.Substitute back into 1 of the new equations to find a 2 nd variable. 6.Substitute both back into one of the original equations to find the 3 rd variable.

Special Situations If you get a false statement (like 0 = -1) when you are trying to solve, the original system has no solution. If you get 0 = 0 when solving, the system has infinitely many solutions.

EXAMPLE 1 Use the elimination method Solve the system. 4x + 2y + 3z = 1 Equation A 2x – 3y + 5z = –14 Equation B 6x – y + 4z = –1 Equation C

EXAMPLE 2 Solve a three-variable system with no solution Solve the system. x + y + z = 3 4x + 4y + 4z = 7 3x – y + 2z = 5

EXAMPLE 3 Solve a three-variable system with many solutions Solve the system. x + y + z = 4 x + y – z = 4 3x + 3y + z = 12

GUIDED PRACTICE for Examples 1, 2 and 3 Solve the system. 1. 3x + y – 2z = 10 6x – 2y + z = –2 x + 4y + 3z = 7 ANSWER (1, 3, –2) 2. x + y – z = 2 2x + 2y – 2z = 6 5x + y – 3z = 8 ANSWER no solution 3. x + y + z = 3 x + y – z = 3 2x + 2y + z = 6 ANSWER Infinitely many solutions

Do #’s 1-9 odd on p. 35 and # 3 on the front Linear Programming Worksheet. Work with a partner quietly. Finish for homework