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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. If the graph of each equation in a system of two variables is a line, then the system is a linear system. 2x – 3y = –2 4x + y = 24

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**Steps for Graphing Paper Calculator Solve both equations for y.**

Plot the y intercept. Plot the slope using rise over run. Solve for y. Enter into y1 and y2. Zoom 2nd Trace #5 Intersect Enter Enter Enter

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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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**Example 2a Real World Connection**

Winning times for the Olympic 400m run have been decreasing more rapidly for women than for men. Use the data in the table to find linear models for women’s and men’s times. Predict the year in which the women’s winning time could equal that of the men, assuming the current trend continues. Year 1968 1972 1976 1980 1984 1988 1992 1996 2000 Men’s Time 43.86 44.66 44.26 44.60 44.27 43.87 43.50 43.49 43.84 Women’s Time 52.03 51.08 49.29 48.88 48.83 48.65 48.25 49.11

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**Example 2 Real World Connection**

Use the models in Example 2 to predict the winning times for the 400m run at the Olympics in 2008 and in 2024. Year 1968 1972 1976 1980 1984 1988 1992 1996 2000 Men’s Time 43.86 44.66 44.26 44.60 44.27 43.87 43.50 43.49 43.84 Women’s Time 52.03 51.08 49.29 48.88 48.83 48.65 48.25 49.11

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**Example: US Life Expectancy at Birth**

Year Men (years) Women (years) 1970 67.1 74.7 1975 68.8 76.6 1980 70.0 77.4 1985 71.1 78.2 1990 71.8 78.8 1995 72.5 78.9 1997 73.6 79.4 Use graphing calculator to find a linear model for the data. Use the model to predict life expectancy in 2015.

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**A system that has a unique solution is an independent system.**

A dependent system does not have a unique solution. An inconsistent system is a system that has no solution.

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**Graphical Solutions of linear Systems in Two Variables**

Intersecting Lines Coinciding Lines Parallel Lines one solution Independent no solution Inconsistent no unique solution Dependent Same slopes and different y intercepts Same slopes and same y intercepts Different slopes and y intercepts

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**Analyzing Graphs Description of Lines How many points of intersection?**

Equal Slopes? (yes or no) Same y-intercepts (yes or no) Intersecting Parallel Coinciding

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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. 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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**Steps for Substitution:**

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 Club Membership Fees 2 months: $100 6 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**

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. 3x + 7y = 15 5x + 2y = -4 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**

Solve the system of inequalities. y ≤ -3/2x + 5 x – 2y < 6 Steps for solving: Graph each linear inequality using boundary lines and then shading. The solution is the intersection of the two shaded regions. Check a point in the intersection.

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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 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. Write a system of equations to represent the cost c for t hours of access in one month for each company. Graph the system from part a. Label each line. For how many hours of use will the costs for the companies be the same? How is this information represented on the graph> 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 hour No 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**

Define the variables. Write a system of inequalities. Graph the system of inequalities on graph paper. Find the coordinates of the vertices of the feasible region. Write a function to be maximized or minimized. Substitute the coordinates of the vertices into the function. 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 case You pay … $24 per case Sell at … $3.50 per can 20 packs per case You pay … $15 per case Sell 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 30 2y 16 500x 200y 500x+200y # of Teams # of Rangers # of Trainees # of Trees Planted

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**Example 2c Real World Connection**

Spruce Maple $30 $40 600 ft2 900 ft2 650 lb/yr 300 lb/yr 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? 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.

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