Chapter 4 Systems of Equations and Problem Solving How are systems of equations solved?

Activation Review Yesterday’s Warm-up

4-1 Systems of Equations in two Variables How do you solve a system of equations in two variables graphically?

Vocabulary Systems of equations: two or more equations using the same variables Linear systems: each equation has two distinct variables to the first degree.

Vocabulary Independent system: one solution Dependent system: many solutions, the same line Consistent: having at least 1 solution Inconsistent system: no solution, parallel lines

Directions: Solve each equation for y Graph each equation State the point of intersection

Examples: x – y = 5 and y + 3 = 2x

Examples: 3x + y = 5 and 15x + 5y = 2

Examples: y = 2x + 3 and -4x + 2y = 6

Examples: x – 2y + 1 = 0 and x + 4y – 6 =0

What limitations do you think are affiliated with this procedure? Method only works well when the solution is integer values. Tomorrow we will investigate other methods

SUMMARIZATION 1. 2.

4-1 Homework PAGE(S): 161 NUMBERS: 2 – 16 even www.phschool.com code age-0775

Activation Review Yesterday’s Warm-up

4-2A Solving Systems of Equations —SUBSTITUTION How do you solve a system of equations in two variables by substitution?

Substitution: example: 4x + 3y = 4 2x – y = 7 1) LOOK FOR A VARIABLE W/O A COEFFICIENT 2) SOLVE FOR THAT VARIABLE 3) SUBSTITUTE THIS NEW VALUE INTO THE OTHER EQUATION example: 4x + 3y = 4 2x – y = 7

Example: 2y + x = 1 3y – 2x = 12

Examples: 5x + 3y = 6 x - y = -1

SUMMARIZATION 1. 2 3

4-2a Homework USING SUBSTITUTION PAGE(S): 166 -167 NUMBERS: 1 – 8 all www.phschool.com code age-0775

Activation Review Yesterday’s warm-up

4-2B and 4-6 Solving Systems of Equations —LINEAR COMBINATION —ELIMINATION METHOD Consistent and Dependent Systems How do you solve a system of equations in two variables by linear combinations? What makes a system dependent, independent, consistent, or inconsistent?

REMEMBER Consistent systems—have at least one solution Inconsistent systems—have no solution Independent systems—have exactly one solution Dependent systems—have an infinite number of solutions

Combination/Elimination 1)LOOK FOR OR CREATE A SET OF OPPOSITES TO CREATE: A) USE THE COEFFICIENT OF THE 1ST WITH THE SECOND AND VICE VERSA B) MAKE SURE THERE WILL BE ONE positive AND ONE negative 2) ADD THE EQUATIONS TOGETHER AND SOLVE 3) SUBSTITUTE IN EITHER EQUATION AND SOLVE FOR THE REMAINING VARIABLE

Example: 4x – 2y = 7 x + 2y = 3

Example: 4x + 3y = 4 2x - y = 7

Example: 3x – 7y = 15 5x + 2y = -4 Vocabulary: consistent solution One value of x and one value of y

Example: 2x - y = 3 -2x + y = -3 Vocabulary: Dependent solution 0 = 0 is TRUE therefore same line

Example: 2x - y = 3 -2x + y = 9 Vocabulary: Inconsistent solution 0 = 12 is FALSE therefore parallel lines Vocabulary: Independent Having one or fewer solutions

Example: 3x + 2y = 7 5x + 7y = 17

SUMMARIZATION 1. 2. 3.

4-2B Homework USING LINEAR COMBINATIONS PAGE(S): 166 -167 NUMBERS: 10 – 22 even www.phschool.com code age-0775

Activation Review Yesterday’s Warm-up

4-3 Using a system of two Equations How do you translate real life problems into systems of equations?

USE ROPES: Read the problem Organize your thoughts in a chart Plan the equations that will work Evaluate the Solution Summarize your findings

Example: The sum of the first number and a second number is -42. The first number minus the second is 52. Find the numbers 1st number x 2nd number y x + y = -42 x - y = 52

Example: Soybean meal is 16% protein and corn meal is 9% protein. How many pounds of each should be mixed together to get a 350 pound mix that is 12% protein? Soybean meal x .16 Corn meal y .09 350 .12 x + y = 350 .16x + .09y = .12 • 350

Example: A total of $1150 was invested part at 12% and part at 11%. The total yield was $133.75. How much was invested at each rate? 12% investment x .12 11% investment y .11 1150 133.75 x + y = 1150 .12x + .11y = 133.75

Example: One day a store sold 45 pens. One kind cost $8.75 the other $9.75. In all, $398.75 was earned. How many of each kind were sold? Pen 1 x 8.75 Pen 2 y 9.75 45 398.75 x + y = 45 8.75x + 9.75y = 398.75

SUMMARIZATION 1. 2.

4-3 Homework PAGE(S): 171 -173 NUMBERS: 4 – 24 by 4’s www.phschool.com code age-0775

Activation Review Yesterday’s Warm-up

4-4 Systems of Equations in three Variables How do you solve a system of equations in three variables? How is it similar to solving a system in two equations?

Find x, y, z 2x + y - z = 5 3x - y + 2z = -1 x - y - z = 0

Find x, y, z 2x - y - 3z = 6 3x + 3y - z = 11 4x + 2y - 3 z = 18

Find x, y, z 4x + 9y = 8 8x + 6z = -1 6y + 6z = -1

Find x, y, z 2x + z = 7 x + 3y + 2z = 5 4x + 2y - 3z = -3

SUMMARIZATION 1. 2.

4-4 Homework PAGE(S): 178 - 179 NUMBERS: 4 – 24 by 4’s www.phschool.com code age-0775

Activation Review Yesterday’s Warm-up

4-5 Using a System of Three Equations How do you translate word problems into a system of three equations?

Example: The sum of three numbers is 105. The third is 11 less than ten times the second. Twice the first is 7 more than three times the second. Find the numbers. 1st number x 2nd number y 3rd number Z x + y + z = 105 z = 10y – 11 2x = 7 + 3y

Example: Sawmills A, B, C can produce 7400 board feet of lumber per day. A and B together can produce 4700 board feet, while B and C together can produce 5200 board feet. How many board feet can each mill produce? Mill A A Mill B B Mill C C A + B + C = 7400 A + B = 4700 B + C = 5200

SUMMARIZATION 1. 2 3

4-5 Homework PAGE(S): 181 - 182 NUMBERS: 4, 8, 12, 16 www.phschool.com code age-0775

Activation Review Yesterday’s Warm-up

4-6b Other Systems of Equations How do you solve other systems of equations?

Find x, y when x is an integer y = x2 y = 3x – 2

Find x, y when x is an integer y = x2 – 2 3x – 2y = 2

Find x, y when x is an integer y + 3 = 2x x2 + 2xy = -1

Find x, y when x is an integer y = x3 + 3x y = 3x – 8

4-6b Homework PAGE(S): worksheet www.phschool.com code age-0775

4-7 Systems of Inequalities How do you solve a system of linear inequalities?

Vocabulary: Feasible region: the area of all possible outcomes

Directions: Solve each equation for y Graph each equation Shade each with lines Shade the intersecting lines a solid color

Examples x – 2y < 6 y ≤ -3/2 x + 5 x – 2y < 6 -2y < 6 – x

y ≤ -2x + 4 x > -3

y < 4 y ≥ |x – 3|

3x + 4y ≥ 12 5x + 6y ≤ 30 1 ≤ x ≤ 3 3x + 4y ≥ 12 4y≥12-3x y ≥ 3 – ¾ x 2 vert lines

SUMMARIZATION 1. 2.

4-7 Homework PAGE(S): 192 NUMBERS: 4 – 32 by 4’s www.phschool.com code age-0775

4. y≤x - 3 x – y ≥ 3 - y ≥ 3 - x y ≤ x - 3

2y - x ≤ 4 2y ≤ 4 + x y ≤ 2 + 1/2x y < x y < x + 0 y> -x + 3

y ≥- 2 x > 1 24. y ≥ -2 y ≥ 2x + 3

y – 2x > 1 y > 1 + 2x y – 2x < 3 y < 3 + 2x 8x + 5y ≤ 40 5y ≤ 40 – 8x y ≤ 8 – 8/5x x + 2y ≤ 8 y ≤ 4 – 1/2x x≥0 y≥0

REVIEW PAGE(S): 200 NUMBERS: all

Activation Review yesterday’s warm-up

4-8 Using Linear Programming EQ: What is linear programming?

Activation x – 2y < 6 y ≤ -3/2 x + 5 x > 0 y > 0

VOCABULARY: Linear programming– identifies minimum or maximum of a given situation Constraints—the linear inequalities that are determined by the problem Objective—the equation that proves the minimum or maximum value.

Directions: Read the problem List the constraints List the objective Graph the inequalities finding the feasible region Solve for the vertices (the points of intersection) Test the vertices in the objective

Example: What values of y maximize P given Constraints: y≥3/2x -3 y ≤-x + 7 x≥0 y≥0 Objective: P = 3x +2y x y P

You are selling cases of mixed nuts and roasted peanuts You are selling cases of mixed nuts and roasted peanuts. You can order no more than a total of 500 cans and packages and spend no more than $600. If both sell equally well, how can you maximize the profit assuming you will sell everything that you buy? items cost Profit per case Mixed Nuts (x items per case) 12 24 Peanuts (y items per case) 15 Totals 500 600 items cost Profit per case Mixed Nuts (x items per case) 12 24 18 Peanuts (y items per case) 15 Totals 500 600 P items cost Profit per case Mixed Nuts (x items per case) 12 Peanuts (y items per case) 15 Totals 500 items cost Profit per case Mixed Nuts (x items per case) Peanuts (y items per case) Totals Constraints: x≥0 y≥0 12x + 20y ≤ 500 y ≤ 25 – 3/5 x 24x + 15y ≤ 600 y ≤ 40 – 8/5 x Objective: P = 18x + 15y x y P

Partner Problem (sample was #8) A florist has to order roses and carnations for Valentine’s Day. The florist needs to decide how many dozen roses and carnations should be ordered to obtain a maximum profit. Roses: The florist’s cost is $20 per dozen, the profit over cost is $20 per dozen. Carnations: The florist’s cost is $5 per dozen, the profit over cost is $8 per dozen. The florist can order no more than 60 dozen flowers. Based on previous years, a minimum of 20 dozen carnations must be ordered. The florist cannot order more than $450 worth of roses and carnations. Find out how many dozen of each the florist should order to max. profit! Cost Total ordered Profit Roses (x) 20 1 Carnations(y) 5 8 Totals 450 60 ? Constraints: x≥0 y≥20 20 x + 5y ≤450 y≤ 90-5x x + y ≤60 y≤ 60-x x y P=20x + 8y 20 20(0)+8(20)=160 60 20(0)+8(60)=480 10 50 20(10)+8(50)=600 17.5=17 (20)17+8(20)=500 Objective: P= 20x + 8y

Sample of what must be handed in for Partner problem

A furniture company makes wooden desks and chairs A furniture company makes wooden desks and chairs. Carpenters and finishers work on each item. On average, the carpenters spend four hours working on each chair and eight hours on each desk. There are enough carpenters for up to 8000 man-hours per week. The finishers spend about two hours on each chair and one hour on each desk. There are enough finishers for a maximum of 1300 man-hours per week. Each chair yields a profit of $150 and each desk $200. Find the maximum amount of profit possible. carpent finisher Profit chairs(x) 4 2 150 desks(y) 8 1 200 Totals 8000 1300 Constraints: x≥0 y≥0 4 x + 8y ≤8000 y≤ 1000-1/2 x 2x + y ≤1300 y≤ 1300-2x Objective: P= 150x + 200y

4-8 Partner PROJECT See worksheet