Chapter 4Systems of Equationsand Problem Solving

How are systems of equations solved?

Activation

• Review Yesterday’s Warm-up

4-1SYSTEMS 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.

Independent system: one solution Dependent system: many solutions,

the same line 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?

4-1HOMEWORK

PAGE(S): 161NUMBERS: 2 – 16 even

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Activation

• Review Yesterday’s Warm-up

4-2ASOLVING SYSTEMS OF EQUATIONS —SUBSTITUTION

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

Substitution:1) LOOK FOR A VARIABLE W/O A

COEFFICIENT2) SOLVE FOR THAT VARIABLE3) SUBSTITUTE THIS NEW VALUE INTO THE

OTHER EQUATION

exampl:e:4x + 3y = 42x – y = 7

• Example:2y + x = 13y – 2x = 12

• Examples:5x + 3y = 6x - y = -1

4-2AHOMEWORK

PAGE(S): 166 -167NUMBERS: 1 – 8 all

USING SUBSTITUTION

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Activation

• Review Yesterday’s warm-up

4-2B AND 4-6 SOLVING SYSTEMS OF EQUATIONS —LINEAR COMBINATION—ELIMINATION METHOD

CONSISTENT AND DEPENDENTSYSTEMS

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

What makes a system dependent, independent, consistent, or inconsistent?

Combination/Elimination1)LOOK FOR OR CREATE A SET OF

OPPOSITESA) TO CREATE USE THE COEFFICIENT OF THE

1ST WITH THE SECOND AND VICE VERSAB) 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

• Example:4x – 2y = 7

x + 2y = 3

• Example:4x + 3y = 42x - y = 7

• Example:3x – 7y = 15 5x + 2y = -4

• Example: 2x - y = 3-2x + y = -3

• Example: 2x - y = 3-2x + y = 9

4-2BHOMEWORK

PAGE(S): 166 -167NUMBERS: 10 – 22 even

USING LINEAR COMBINATIONS

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Activation

• Review Yesterday’s Warm-up

4-3USING 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

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

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?

Type 1 x 8.75

Type 2 y 9.75

x + y = 458.75x + 9.75y = 398.75

4-3HOMEWORK

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

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Activation

• Review Yesterday’s Warm-up

4-4SYSTEMS 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, z2x + y - z = 53x - y + 2z = -1 x - y - z = 0

Find x, y, z2x - y + z = 4 x + 3y - z = 114x + y - z = 14

Find x, y, z2x + z = 7 x + 3y + 2z = 54x + 2y - 3z = -3

4-4HOMEWORK

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

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Activation

• Review Yesterday’s Warm-up

4-5USING 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 x

Mill B y

Mill C z

x + y + z = 7400 x + y = 4700 y + z = 5200

4-5HOMEWORK

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

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Activation

• Review Yesterday’s Warm-up

4-7SYSTEMS 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

y ≤ -2x + 4 x > -3

y < 4y ≥ |x – 3|

3x + 4y ≥ 12

5x + 6y ≤ 301 ≤ x ≤ 3

4-7HOMEWORK

PAGE(S): 192NUMBERS: 4 – 32 by 4’s

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REVIEW

PAGE(S): 200 NUMBERS: all

Activation

• Review yesterday’s warm-up

4-8USING LINEAR PROGRAMMING

EQ: What is linear programming?

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 givenConstraints: y≥3/2x -3 y ≤-x + 7 x≥0 y≥0Objective:

P = 3x +2y

x y P

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

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

Sample of what must be handed in for Partner

problem

4-8PARTNER PROJECT

See worksheet