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**Chapter 4 Systems of Equations and Problem Solving**

How are systems of equations solved?

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Activation Review Yesterday’s Warm-up

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**4-1 Systems of Equations in two Variables**

How do you solve a system of equations in two variables graphically?

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Vocabulary Systems of equations: two or more equations using the same variables Linear systems: each equation has two distinct variables to the first degree. Independent system: one solution Dependent system: many solutions, the same line Inconsistent system: no solution, parallel lines

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**Directions: Solve each equation for y Graph each equation**

State the point of intersection

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Examples: x – y = 5 and y + 3 = 2x

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

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

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

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**What limitations do you think are affiliated with this procedure?**

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**4-1 Homework PAGE(S): 161 NUMBERS: 2 – 16 even**

code age-0775

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Activation Review Yesterday’s Warm-up

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**4-2A Solving Systems of Equations —SUBSTITUTION**

How do you solve a system of equations in two variables by substitution?

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**Substitution: exampl:e: 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 exampl:e: 4x + 3y = 4 2x – y = 7

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Example: 2y + x = 1 3y – 2x = 12

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

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**4-2a Homework USING SUBSTITUTION PAGE(S): 166 -167 NUMBERS: 1 – 8 all**

code age-0775

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Activation Review Yesterday’s warm-up

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4-2B and 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?

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**Combination/Elimination**

1)LOOK FOR OR CREATE A SET OF OPPOSITES A) TO CREATE USE THE COEFFICIENT OF THE 1ST WITH THE SECOND AND VICE VERSA B) MAKE SURE THERE WILL BE ONE + & ONE – 2) ADD THE EQUATIONS TOGETHER AND SOLVE 3) SUSTITUTE IN EITHER EQUATION AND SOLVE FOR THE REMAINING VARIABLE

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Example: 4x – 2y = 7 x + 2y = 3

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

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Example: 3x – 7y = 15 5x + 2y = -4

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

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Example: 2x - y = 3 -2x + y = 9

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**4-2B Homework USING LINEAR COMBINATIONS PAGE(S): 166 -167**

NUMBERS: 10 – 22 even code age-0775

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Activation Review Yesterday’s Warm-up

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**4-3 Using a system of two Equations**

How do you translate real life problems into systems of equations?

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

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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

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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 x y = 350 .16x + .09y = .12 • 350

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

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

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**4-3 Homework PAGE(S): 171 -173 NUMBERS: 4 – 24 by 4’s**

code age-0775

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Activation Review Yesterday’s Warm-up

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**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?

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

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Find x, y, z 2x - y + z = 4 x + 3y - z = 11 4x + y - z = 14

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

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**4-4 Homework PAGE(S): 178 - 179 NUMBERS: 4 – 24 by 4’s**

code age-0775

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Activation Review Yesterday’s Warm-up

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**4-5 Using a System of Three Equations**

How do you translate word problems into a system of three equations?

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Example: The sum of three numbers is 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 = y

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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 x Mill B y Mill C z x + y + z = 7400 x + y = 4700 y + z = 5200

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**4-5 Homework PAGE(S): 181 - 182 NUMBERS: 4, 8, 12, 16**

code age-0775

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Activation Review Yesterday’s Warm-up

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**4-7 Systems of Inequalities**

How do you solve a system of linear inequalities?

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Vocabulary: Feasible region: the area of all possible outcomes

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**Directions: Solve each equation for y Graph each equation**

Shade each with lines Shade the intersecting lines a solid color

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Examples x – 2y < 6 y ≤ -3/2 x + 5

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y ≤ -2x + 4 x > -3

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y < 4 y ≥ |x – 3|

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3x + 4y ≥ 12 5x + 6y ≤ 30 1 ≤ x ≤ 3

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**4-7 Homework PAGE(S): 192 NUMBERS: 4 – 32 by 4’s**

code age-0775

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REVIEW PAGE(S): 200 NUMBERS: all

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Activation Review yesterday’s warm-up

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**4-8 Using Linear Programming**

EQ: What is linear programming?

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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.

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**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

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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

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**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? x y P

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**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 x y P=20x + 8y

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**Sample of what must be handed in for Partner problem**

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4-8 Partner PROJECT See worksheet

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Learning Objectives for Section 4.1

Learning Objectives for Section 4.1

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