Chapter 3

Linear Systems

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In Chapter 3, You Will…

Learn to solve systems of equations and inequalities in two variables algebraically and by graphing.

Learn to graph points and equations in three dimensions.

Learn to solve systems of equations in three variables.

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3-1 Graphing Systems of Equations

What you’ll learn … To solve a system by graphing

2.10 Use systems of two or more equations or inequalities to model and solve problems; justify results. Solve using tables, graphs, matrix operations, and algebraic properties.

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A system of equations is a set of two or more equations that use the same variables. The solution to a system of two variables is the point that makes both statement true. In terms of the graph, the solution is the point where the 2 lines intersect.

2x – 3y = –24x +  y = 24

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Steps for Graphing

Solve both equations for y. Plot the y intercept. Plot the slope using rise over run.

Paper

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Example 1 Solving by Graphing

Solve the system by graphing.

x + 2y = -7 2x – 3y = 0

• 2x +y = 5• -x +y = 2

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Intersecting Lines Parallel Lines Coinciding Lines

one solutionIndependent

no solutionInconsistent

no unique solutionDependent

Graphical Solutions of linear Systems in Two Variables

Different slopes and y intercepts

Same slopes and different y intercepts

Same slopes and same y intercepts

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Example 3 Classifying Systems Without Graphing

Classify the system without graphing.

• 3x + y = 5• 15x + 5y = 2

y = 2x +3 -4x + 2y = 6

x – y = 5 y + 3 = 2x

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3-2 Solving Systems Algebraically

What you’ll learn … To solve a system by substitution. To solve a system by substitution.

2.10 Use systems of two or more equations or inequalities to model and solve problems; justify results. Solve using tables, graphs, matrix operations, and algebraic properties.

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Substitution Method:

Steps for Substitution: Solve for x or y in one

of the equations. Substitute into the

second equation. Solve for the variable in

the second equation. Substitute that value

into one of the equations to solve for the other unknown variable.

A method of solving a system of equations by replacing one variable with an equivalent expression containing the other variable.

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Example 1: Solving by Substitution

Solve the system by substitution.

4x +3y = 4

2x – y =7

Solve the system by substitution.

2x – 3y = 6

x + y = -12

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Example 2a: Real World Connection

Refer to the photo at the left. The cost of membership in a health club includes a monthly charge and a one time initiation fee. Find the monthly charge and the initiation fee.

Health ClubMembership Fees

2 months: $1006 months $200

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Example 2b: Real World Connection

You can buy CDs at a local store for $15.49 each. You can buy them at an online store for $13.99 each plus $6 for shipping. Solve a system of equations to find the number of CDs that you can buy for the same amount at the two stores.

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Elimination Method:

Steps for Elimination Put both equations into

standard form. Eliminate by getting

opposite coefficients. Add the two equations,

solve. Substitute value into

equation for unknown variable.

A method of solving a system of equations. You add or subtract the equations to eliminate a variable.

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Example 3: Solving by Elimination

Use the elimination method to solve the system.

4x – 2y = 7

x + 2y = 3

Use the elimination method to solve the system.

4x + 9y = 1

4x + 6y = -2

Use the elimination method to solve the system.

3x + 7y = 15

5x + 2y = -4

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Example 4: Solving by Elimination

Use the elimination method to solve the system.

3x + 7y = 15

5x + 2y = -4

To make two terms additive inverses, you may need to multiply one or both equations in a system by a nonzero number. In doing so, you create a system equivalent to the original one. Equivalent systems are systems that have the same solution (s).

Use the elimination method to solve the system.

2m + 4n = -4

3m + 5n = -3

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Example 5: Solving a System Without a Unique Solution

Solve the system using substitution or elimination method.

-3x + 5y = 7

6x - 10y = -14

Solve the system using substitution or elimination method.

-2x + 4y = 6

-3x + 6y = 8

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3-3 Systems of Inequalities

What you’ll learn … To solve a system of linear inequalities.

2.10 Use systems of two or more equations or inequalities to model and solve problems; justify results. Solve using tables, graphs, matrix operations, and algebraic properties.

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Example 1a: Solving a System of Inequalities

Steps for solving:

1. Graph each linear inequality using boundary lines and then shading.

2. The solution is the intersection of the two shaded regions.

3. Check a point in the intersection.

Solve the system of inequalities.

y ≤ -3/2x + 5

x – 2y < 6

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Example 1b: Solving a System of Inequalities

Solve the system of inequalities.

y ≤ -2x + 4

x > -3

y ≤ 3x - 6

y > -4x + 2

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Example 2a Real World Connection

An entrance exam has two parts, a verbal part and a math part. You can score a maximum total of 1600 points. For admission, the school of your choice requires a math score of at least 600. Write and solve a system of inequalities to model scores that meet the school’s requirements.

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Example 2b Real World Connection

Another school requires a math score of at least 550 points and a total score of at least 1100 points. You can score up to 800 points on each part. Write and solve a system of inequalities to model scores that meet the school’s requirements.

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Example 3: Solving a Linear Absolute Value System

Solve the system of inequalities.

y < 4

y ≥ x - 3

y ≥ -2x + 4

y ≤ x - 4

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Application A youth group with 26 members is going skiing.

Each of the five chaperones will drive a van or a sedan. The vans can seat seven people, and the sedans can seat five people. How many of each type of vehicle could transport all 31 people to the ski area in one trip ?

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Application

A boat can travel 24 miles in 3 hours when traveling with the current. Against the same current, it can travel only 16 miles in 4 hours. Find the rate of the current and the rate of the boat in still water.

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Application

In a mayoral election, the incumbent received 25% more votes than the opponent. Altogether, 5175 votes were cast for the two candidates. How many votes did the incumbent mayor receive?

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Application The ads at the left show the

costs of Internet access for two companies.

a. Write a system of equations to represent the cost c for t hours of access in one month for each company.

b. Graph the system from part a. Label each line.

c. For how many hours of use will the costs for the companies be the same? How is this information represented on the graph>

d. If you use the Internet about 20 hours each month, which company should you choose? Explain how you reached an answer.

$2.25 per hour$9.95 base fee (per month)

$2.95 per hourNo base fee

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Application A bookstore took in

$167 on the sale of 5 copies of a new cookbook and 3 copies of a new novel. The next day it took in $89 on the sale of 3 copies of the cookbook and 1 copy of the novel. What was sale price of each book?

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3-4 Linear Programming

What you’ll learn … To find maximum and minimum values. To solve problems with linear programming.

2.10 Use systems of two or more equations or inequalities to model and solve problems; justify results. Solve using tables, graphs, matrix operations, and algebraic properties.

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Linear programming is a technique that identifies the minimum or maximum value of some quantity. This quantity is modeled with an objective function. Limits on the variables in the objective function are constraints, written as linear inequalities.

These constraints form the system of

inequalities at the right. The blue region in the graph, the feasible region, contains all points that satisfy all the constraints.

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Steps to Solve Linear Programming Problem

1. Define the variables.2. Write a system of inequalities.3. Graph the system of inequalities on graph paper.4. Find the coordinates of the vertices of the

feasible region.5. Write a function to be maximized or minimized.6. Substitute the coordinates of the vertices into the

function.7. Select the greatest or least result. Answer the

problem.

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Example 2 Find the values of x and y that maximize or

minimize the objective function for each graph.

D. (0, 500)

C. (400, 300)

B. (600, 0)A. (0, 0)

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Example 6

Graph each system of constraints. Name all vertices. Then find the values of x and y that maximize or minimize the objective function.

x + y ≤ 8

2x + y ≤ 10

x ≥ 0

y ≥ 0

Maximize for N = 100x + 40y

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Example 9

Graph each system of constraints. Name all vertices. Then find the values of x and y that maximize or minimize the objective function.

2 ≤ x ≤ 6

1 ≤ y ≤ 5

x + y ≤ 8

Maximize for P = 3x + 2y

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Example 2a Real World Connection

Suppose 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. How can you maximize your profit? How much is the maximum profit?

12 cans per caseYou pay … $24 per caseSell at … $3.50 per can

20 packs per caseYou pay … $15 per caseSell at … $1.50 per pack

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Example 2b Real World Connection

Teams chosen from 30 forest rangers and 16 trainees are planting trees. An experienced team consisting of two rangers can plant 500 trees per week. A training team consisting of one ranger and two trainees can plant 200 trees per week.

Experienced Teams

Training Teams

Total

X Y X + y

2x Y 30

0 2y 16

500x 200y 500x+200y

# of Teams

# of Rangers

# of Trainees

# of Trees Planted

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Example 2c Real World Connection

Trees in urban areas help keep air fresh by absorbing carbon dioxide. A city has $2100 to spend on planting spruce and maple trees. The land available for planting is 45,000 ft2. How many of each tree should the city plant to maximize carbon dioxide absorption?

Spruce Maple

$30 $40

600 ft2 900 ft2

650 lb/yr 300 lb/yr

Planting Cost

Area Required

CO2 Absorption

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Example 2d Real World Connection

A biologist is developing two new strains of bacteria. Each sample of Type I bacteria produces four new viable bacteria, and each sample of Type II produces three new viable bacteria. Altogether, at least 240 new viable bacteria must be produced. At least 30, bit not more than 60, of the original samples must be Type I. Not more than 70 of the samples can be Type II. A sample of Type I costs $5 and a sample of Type II costs $7. How many samples of each should be used to minimize cost?

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Example 2e Real World Connection

Baking a tray of corn muffins takes 4 cups of milk and 3 cups of wheat flour. A tray of bran muffins takes 2 cups of milk and 3 cups of wheat flour. A baker has 16 cups of milk and 15 cups of wheat flour. He makes $3 profit per tray of corn muffins and $2 profit per tray of bran muffins. How many trays of each type of muffins should the baker make to maximize his profits?

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In Chapter 3, You Should Have…

Learned to solve systems of equations and inequalities in two variables algebraically and by graphing.

Learned to graph points and equations in three dimensions.

Learned to solve systems of equations in three variables.