solving systems of linear equations what’s in the mix? 3

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Unit 3 • Extensions of Linear Concepts 179

ACTIVITY

My Notes

ACTIVITY

SUGGESTED LEARNING STRATEGIES: Shared Reading, Discussion Group, Create Representations

Systems of equations are useful for solving a variety of problems. Chemists form solutions by mixing liquids. A saline solution can be formed by dissolving salt in water or mixing together other saline solutions.

EXAMPLE 1

Noah is given two beakers of saline solution in chemistry class. One contains a 3% saline solution and the other an 8% saline solution. How much of each type of solution will Noah need to mix to create 150 mL of a 5% solution?Step 1: To solve this problem, write and solve a system of linear equations.

Let x = number of mL of 3% solution.Let y = number of mL of 8% solution.

Step 2: Write one equation based on the amounts of liquid being mixed. Write another equation on the amount of saline in the fi nal solution.

x + y = 150 Th e amount of the mixture is 150 mL. 0.03x + 0.08y = 0.05(150) Th e amount of saline in the mixture is 5%.Step 3: To solve this system of equations by elimination, decide to

eliminate the x variable.-3(x + y) = -3(150) Multiply the fi rst equation by -3.

100(0.03x + 0.08y) = 100(7.5) Multiply the second equation by 100 to remove decimals.

-3x - 3y = -450 3x + 8y = 750 Add the two equations to eliminate x.

5y = 300 Solve for y.y = 60

Step 4: Find the value of the eliminated variable x by using one of the original equations.

x + y = 150 x + 60 = 150 Substitute 60 for y. x = 90 Subtract 60 from both sides.Step 5: Check your answers by substituting into the original second

equation. 0.03x + 0.08y = 0.05(150)0.03(90) + 0.08(60) ? 0.05(150) Substitute 90 for x and 60 for y. 2.7 + 4.8 ? 7.5 7.5 = 7.5 checkSolution: Noah needs 90 mL of 3% solution mixed with 60 mL of 8%

solution to make 150 mL of the 5% solution.

CONNECT TO SCIENCESCIENCE

In chemistry, a solution refers to a mixture of two or more substances. In mathematics, a solution can refer to a method for solving a problem or it can be the answer to a problem.

Solving Systems of Linear EquationsWhat’s in the Mix? 3.6

ACADEMIC VOCABULARY

The elimination method for solving a system of two linear equations involves eliminating one variable. To eliminate one variable, multiply each equation in the system by an appropriate number so that the terms for one of the variables will combine to zero when the equations are added. Then substitute the value of the known variable to fi nd the value of the unknown variable. The ordered pair is the solution of the system.

SB_A1_3-6_SE.indd 179SB_A1_3-6_SE.indd 179 4/4/09 11:39:09 PM4/4/09 11:39:09 PM

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180 SpringBoard® Mathematics with Meaning™ Algebra 1

My Notes

Solving Systems of Linear Equations ACTIVITY 3.6continued What’s in the Mix?What’s in the Mix?

TRY THESE A

a. Mary has $25,000 to invest. She decides to invest part of that amount at 3% and part at 5% interest for one year. Th e amount of interest she earns for both investments is $1100. How much was invested at each rate?

b. Solve the system using the elimination method: 7x + 5y = -14x - y = -16

c. Sylvia wants to mix 100 pounds of Breakfast Blend coff ee that will sell for $25 per pound. She is using two types of coff ee to create the mixture. Kona coff ee sells for $51 per pound and Columbian coff ee sells for $11 per pound. How many pounds of each type of coff ee should she use?

EXAMPLE 2

Solve the system using the linear combination method: 4x - 5y = 303x + 4y = 7

Step 1: To solve this system of equations by linear combination, decide to eliminate the y variable.

Original system Multiply the fi rst Add the two equations equation by 4. to eliminate y. Multiply the second equation by 5.4x - 5y = 30 4(4x - 5y) = 4(30) 16x - 20y = 1203x + 4y = 7 5(3x + 4y) = 5(7) 15x + 20y = 35 31x = 155 Solve for x. x = 5Step 2: Find y by substituting the value of x into the fi rst

original equation. 4x - 5y = 30 4(5) - 5y = 30 Substitute 5 for x. 20 - 5y = 30 -5y = 10 y = -2

Step 3: Check (5, -2) in the second equation 3x + 4y = 7. 3x + 4y = 73(5) + 4(-2) ? 7

15 - 8 ? 7 7 = 7 check

Solution: Th e solution is (5, -2).

Recall the formula for simple interest,

I = prt,whereI = interest earnedp = principal amountr = ratet = time in years

The elimination method for solving a system of linear equations is also called the linear combination method.

MATH TERMS

SUGGESTED LEARNING STRATEGIES: Shared Reading, Discussion Group, Self/Peer Revision


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My Notes

ACTIVITY 3.6continued

Solving Systems of Linear Equations What’s in the Mix?What’s in the Mix?

TRY THESE B

a. Solve the system of equations: 3x - 2y = -212x + 5y = 5

b. Solve the system of equations: 7x + 5y = 94x - 3y = 11

c. Phuong has $2.00 in nickels and dimes in his bank. Th e number of dimes is fi ve more than twice as many nickels. How many of each type of coin are in his bank?

Another method for solving systems of equations involves substitution. Th is method is similar to the algebraic one you learned in Activity 3.4.

EXAMPLE 3

For the Valentine’s Day Dance, tickets for couples cost $12 and tickets for individuals cost $8. Suppose 250 students attended the dance and $1580 was collected from ticket sales. How many of each type of ticket was sold?

Step 1: Let x = number of couples and y = number of single people.Step 2: Write one equation to represent the number of people attending.

Write another equation to represent the money collected. 2x + y = 250 Th e number of attendees is 250. 12x + 8y = 1580 Th e total ticket sales is $1580.

Step 3: Use substitution to solve this system. 2x + y = 250 Solve the fi rst equation for y.

y = 250 - 2x

12x + 8(250 - 2x) = 1580 Substitute for y in the second equation. 12x + 2000 - 16x = 1580 Solve for x.

-4x = -420x = 105

Step 4: Substitute the value of x into one of the original equations to fi nd y.

2x + y = 250 2(105) + y = 250 Substitute 105 for x.

210 + y = 250 y = 40Solution: For the dance, 105 couples’ tickets and 40 singles’ tickets

were sold.

SUGGESTED LEARNING STRATEGIES: Self/Peer Revision, Discussion Group, Create Representations

ACADEMIC VOCABULARY

The substitution method for solving systems of equations involves solving one of the equations for a variable, then substituting the value of that variable into other equation to form a single equation in one variable. The equation is then solved, and substitution is used to fi nd the value of the unknown variable.


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My Notes


TRY THESE C

a. Solve the system x + 2y = 8 and 3x - 4y = 4 by substitution.

b. Solve the system of equations: 5x - 2y = 03x + y = -1

c. Patty and Toby live 345 miles apart. Th ey decide to drive to meet one another. Patty leaves at noon traveling at an average rate of 45 mph and Toby leaves at 3:00 pm traveling at an average speed of 60 mph. At what time will they meet?

When a system of two linear equations in two variables is solved, three possible relationships can occur.

• Two distinct lines that intersect with one ordered pair as the solution.

• Two distinct lines that do not intersect because the lines are parallel and have no solution.

• Two lines that are coincident produce the same solution set—an infi nite set of ordered pairs that satisfy both equations

Systems of linear equations are classifi ed by the relationships of their lines. Systems that produce two distinct lines when graphed are said to be independent. Systems that have no solution are said to be inconsistent.

Th e three systems in the chart represent each of the possible relationships described above.

Relationship of Lines Sketch Classifi cation

Two Intersecting Lines Independent and Consistent

Two Parallel Lines Independent and Inconsistent

Two Coincident Lines Dependent and Consistent

Coincident lines occupy the same space or location in the plane and pass through the same set of ordered pairs.

MATH TERMS

SUGGESTED LEARNING STRATEGIES: Discussion Group, Summarize/Paraphrase/Retell


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My Notes



SUGGESTED LEARNING STRATEGIES: Mark the Text, Quickwrite

1. For each system below, complete the table with the information requested.

Th e Nature of Solutions to a System of Two Linear EquationsEquations in Standard Form: 2x + y = 2 2x + y = 2 2x + y = 2

6x + 3y = 6 x + y = 3 4x + 2y = -4Graph each system:

Write the Number of Solutions:

Write the Relationship of the Lines:

Solve Algebraically:

Write the Equations in Slope-Intercept Form:

Compare the Slopes and y-intercepts:

Classify the System:

x

y

45

321

–4–5 –3 –2 –1 1 2 3 4 5–1–2–3–4–5

x

y

45

321

–4–5 –3 –2 –1 1 2 3 4 5–1–2–3–4–5

x

y

45

321

–4–5 –3 –2 –1 1 2 3 4 5–1–2–3–4–5


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My Notes


SUGGESTED LEARNING STRATEGIES: Self/Peer Revision, Discussion Group

TRY THESE D

For each system below: i. Tell how many solutions the system has. ii. Describe the graph.iii. Classify the system.

a. 2x - 2y = 6 b. y = 1.5x + 5y - x = -3 3x - 2y = 10

c. y = 2 __ 3 x + 1 d. 3x + 4y = 14x - 6y = - 6 2x - 5y = 16

Sometimes a situation has more than two pieces of information. For these more complex problems, you may need to solve equations that contain three variables. Th e solution will be the ordered triple (x, y, z).

EXAMPLE 4

Solve the system by elimination. x + 2y - z = -3-x - 3y + 2z = 7

-2x + y + z = -2Step 1: First, eliminate the same variable from two pairs of equations.

Next use the resulting system of two-variable equations to fi nd the values of the two variables. Th en substitute those values into an original equation to fi nd the value of the third variable.

Step 2: Use the fi rst and second equations to eliminate x. Add the equations.

x + 2y - z = -3 -x - 3y + 2z = 7

-y + z = 4

Step 3: Use the fi rst and third equations to eliminate x again. Multiply the fi rst equation by 2 so that the x-terms add to zero.

2( x + 2y - z) = 2(-3) 2x + 4y - 2z = -6-2x + y + z = -2 2x + y - z = -2

5y - z = -8

(continued on next page)

You can use either elimination or substitution to solve a system of three equations in three variables.

Use elimination if the terms easily add to 0. Use substitution if one equation has only one variable on one side, such as x = 2y + z.


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My Notes



EXAMPLE 4 (continued)

Step 4: Use the two equations you found to write a new system with two variables. Add the equations to fi nd the value of y, because the z-terms add to zero.

-y + z = 4 5y - z = -84y = -4

y = -1Step 5: Substitute the y-value into one of these equations to fi nd z.

-y + z = 4 -(-1) + z = 4 1 + z = 4

z = 3Step 6: Substitute the y- and z-values into one of the original

equations to fi nd x.x + 2y - z = -3x + 2(-1) - 3 = -3x - 2 - 3 = -3x - 5 = -3x = 2

Solution: Th e solution of the system is (2, -1, 3).

TRY THESE E

a. Check the solution in Example 4 by substituting (2, -1, 3) into all three of the original equations.

b. Solve the system by elimination. 2x + 2y - z = -10x - y + z = 4

2x + y + z = 7Tickets for the Spring Festival cost $5 each. Children under 5 years pay $3 and senior citizens pay $4. A total of $870 was collected from ticket sales, and there were four times as many regular tickets as senior tickets sold. Th ere were 200 people who attended the festival. Th e system below represents the relationships in the problem.

x + y + z = 2003x + 5y + 4z = 870 y = 4z

c. What does x represent in the problem? What does y represent? What does z represent?

SUGGESTED LEARNING STRATEGIES: Discussion Group, Create Representations


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My Notes


SUGGESTED LEARNING STRATEGIES: Create Representations, Quickwrite, Self/Peer Revisions

TRY THESE E (continued)

d. You can use substitution to solve the system above. Substitute 4z for y in the fi rst two equations. What new system in two variables do you get?

e. Solve that two-variable system. What values of y and z did you fi nd?

f. Substitute your y- and z-values into an original equation. What value of x did you fi nd?

g. How many of each type of ticket was sold?

Write your answers on notebook paper. Show your work.

CHECK YOUR UNDERSTANDING

1. Solve y = - 1 __ 2 x + 5 and 3x - y = 2 by

substitution. Classify the solutions. 2. Solve 5x + 6y = 8 and 2x - 3y = 5 using

the elimination method. 3. Solve the system 3x - 4y = 8 and y = 3 __ 4 x - 2 using any method. Classify the solutions.4. Approximate the point of intersection for

the system of linear equations graphed below. Verify that the selected point is a solution for the system.

x

8

10

12

6

4

2

–4 –2 2

12

4 6 8 10 12–2

–4

y

y = – x + 9

y = 2x – 1

5. Jillian wants to create a 20% solution of ethanol. She has 300 mL of a 4% solution and pure 100% ethanol. How much pure ethanol should she mix with the 4% solution and how much of the 20% solution will be produced aft er the mixing is completed?

6. Find the solution of the system y = -

2 __ 5 x + 1 and 2x + 5y = 3 by any method. Classify the solutions.

7. Solve the system. x - y + z = -1 2x + y + z = 5 x - 2y - 3z = -13 8. MATHEMATICAL

R E F L E C T I O N Consider the diff erent methods that have been

studied in Tale of Two Truckers and in this activity to write about when it is advantageous to use one method of solution instead of another. Be sure to consider and comment on each of the methods:

Substitution Using Tables

Graphing Elimination


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