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Teacher Copy
Lesson 3-2
Solving Systems of Three Equations in Three Variables
Plan
Pacing: 1 class period
Chunking the Lesson
Example A #1 Example B
p. 36
Example C #2
Check Your Understanding
Lesson Practice
Teach
Bell-Ringer Activity
Students should recall that an absolute value of a number is its distance from zero on a number line.
Have students evaluate the following:
1. |6| [6]
2. |–6| [6]
Then have students solve the following equation.
3. |x|= 6 [x = 6 or x = –6]
Example A Marking the Text, Interactive Word Wall
©
2014
College
Board.
All rights
reserved.
Point
out the
Math Tip
to reinforce
why two
solutions exist. Work through the solutions to the equation algebraically. Remind students that
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solutions to an equation make the equation a true statement. This mathematical understanding is necessary for students to be able to check
their results.
Developing Math Language
An absolute value equation is an equation involving an absolute value of an expression containing a variable. Just like when solving
algebraic equations without absolute value bars, the goal is to isolate the variable. In this case, isolate the absolute value bars because they
contain the variable. It should be emphasized that when solving absolute value equations, students must think of two cases, as there are two
numbers that have a specific distance from zero on a number line.
1 Identify a Subtask, Quickwrite
When solving absolute value equations, students may not see the purpose in creating two equations. Reviewing the definition of the absolute
value function as a piecewise-defined function with two rules may enable students to see the reason why two equations are necessary.
Have students look back at Try These A, parts c and d. Have volunteers construct a graph of the two piecewise-defined functions used to
write each equation and then discuss how the solution set is represented by the graph.
Example B Marking the Text, Simplify the Problem, Critique Reasoning, Group Presentation
Start with emphasizing the word vary in the Example, discussing what it means when something varies. You may wish to present a simpler
example such as: The average cost of a pound of coffee is $8. However, the cost sometimes varies by $1. This means that the coffee could
cost as little as $7 per pound or as much as $9 per pound. Now have students work in small groups to examine and solve Example B by
implementing an absolute value equation. Additionally, ask them to take the problem a step further and graph its solution on a number line.
Have groups present their findings to the class.
ELL Support
For those students for whom English is a second language, explain that the word varies in mathematics means changes. There are different
ways of thinking about how values can vary. Values can vary upward or downward, less than or greater than, in a positive direction or a
negative direction, and so on. However, the importance comes in realizing that there are two different directions, regardless of how you think
of it.
Also address the word extremes as it pertains to mathematics. An extreme value is a maximum value if it is the largest possible amount
(greatest value), and an extreme value is a minimum value if it is the smallest possible amount (least value).
Developing Math Language
An absolute value inequality is basically the same as an absolute value equation, except that the equal sign is now an inequality symbol: <,
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>, ≤, ≥, or ≠. It still involves an absolute value expression that contains a variable, just like before. Use graphs on a number line of the
solutions of simple equations and inequalities and absolute value equations and inequalities to show how these are all related.
Example C Simplify the Problem, Debriefing
Before addressing Example C, discuss the following: Inequalities with |A| > b, where b is a positive number, are known as disjunctions
and are written as A < –b or A > b.
For example, |x| > 5 means the value of the variable x is more than 5 units away from the origin (zero) on a number line. The solution is x
< – 5 or x > 5.
See graph A.
This also holds true for |A| ≥ b.
Inequalities with |A| < b, where b is a positive number, are known as conjunctions and are written as –b < A < b, or as –b < A and A < b.
For example: |x| < 5; this means the value of the variable x is less than 5 units away from the origin (zero) on a number line. The solution
is –5 < x < 5.
See graph B.
This also holds true for |A| ≤ b.
Students can apply these generalizations to Example C. Point out that they should proceed to solve these just as they would an algebraic
equation, except in two parts, as shown above. After they have some time to work through parts a and b, discuss the solutions with the
whole class.
Teacher to Teacher
Another method for solving inequalities relies on the geometric definition of absolute value |x – a| as the distance from x to a. Here’s how
you can solve the inequality in the example:
Thus, the solution set is all values of x whose distance from is greater than . The solution can be represented on a number line and written
as x < –4 or x > 1.
2 Quickwrite, Self Revision/Peer Revision, Debriefing
Use the investigation regarding the restriction c > 0 as an opportunity to discuss the need to identify impossible situations involving
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inequalities.
Check Your Understanding
Debrief students’ answers to these items to ensure that they understand concepts related to absolute value equations. Have groups of
students present their solutions to Item 4.
Assess
Students’ answers to Lesson Practice problems will provide you with a formative assessment of their understanding of the lesson
concepts and their ability to apply their learning.
See the Activity Practice for additional problems for this lesson. You may assign the problems here or use them as a culmination for the
activity.
Adapt
Check students’ answers to the Lesson Practice to ensure that they understand basic concepts related to writing and solving absolute
value equations and inequalities and graphing the solutions of absolute value equations and inequalities. If students are still having
difficulty, review the process of rewriting an absolute value equation or inequality as two equations or inequalities.
Plan
Pacing: 1 class period
Chunking the Lesson
Example A Example B
#1 #2–5 #6–7
Check Your Understanding
Lesson Practice
Teach
Bell-Ringer Activity
Write the following on the board:
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If z = 15, find the ordered pair (x, y) that satisfies the system of equations:
x + y + z = −1
2x − 2y + 3z = 8
Discuss students’ answers and have them share how they solved the problem.
Introduction Close Reading, Vocabulary Organizer
Solving this contextual problem is not part of this activity. The problem is used simply to illustrate one type of situation that can be
represented by three variables in a system of linear equations.
Developing Math Language
Many students have trouble visualizing a three-dimensional coordinate system when it is represented with a two-dimensional
drawing. Help students understand the three-dimensional coordinate system by using the physical classroom as a model. The floor of
the classroom can be thought of as Quadrant I of the xy-plane and one corner of the room can be thought of as the positive z-axis.
Adjoining rooms on the same floor can be visualized to extend the xy-plane, and rooms below your classroom can be visualized to
represent the negative z-axis.
Example A Note Taking
Point out that solving systems in three variables is similar to solving systems in two variables once a variable term has been
eliminated. In this case, after the x-terms are eliminated from the three original equations, students will be solving a system in two
variables—y and z.
Differentiating Instruction
For students who need a challenge beyond Example 3, assign the problem of finding the monetary value of the souvenir nuggets on
the previous page. The correct solution is g = $0.50, s = $0.35, b = $0.25.
Example B Work Backward
Students often have trouble locating the source of the error when they do not arrive at the correct solution when solving a system of
equations in three variables. Encourage them to write notes about what they are doing and label equations so that they can more
easily retrace their steps when checking their work.
Universal Access
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For students who are struggling with solving systems in three variables in which all three variables are present in all three
equations, consider beginning with the following two systems.
1. x + y + z = 4; y + z = 12; y = 6 [(4, 6, 6)]
2. x + y + z = 10; x − y = −1; x + y = 5 [(2, 3, 5)]
Differentiating Instruction
Challenge students to write a system of three equations in three variables that has the solution (3, −1, 5). Point out that there are an
infinite number of systems with this solution. If necessary, provide the hint that students need to work backward.
1 Think-Pair-Share
Have students trade flowcharts with a partner and then follow that flowchart to solve the system 2x − y − z = 10; 3x − 2y − 2z = 7;
x − 3y − 2z = 10. Students can critique their partner’s flowcharts based on their effectiveness for solving the given system.
Teacher to Teacher
No two students seem to follow the exact same steps when solving a system of equations in three variables, which can make it
difficult when students ask you for help. Have students verbally walk you through what they have done so far rather than
beginning the solution process from scratch with them. It is important to model to students that understanding others’
mathematical thinking is a good practice.
Differentiating Instruction
Tell students that an equation in three variables represents a plane in three-dimensional space. Have students draw diagrams
showing the possible ways to position three planes in three-dimensional space and then label the diagrams with the terms
consistent, inconsistent, dependent, and independent.
2–5 Create Representations, Guess and Check, Sharing and Responding
After students have written their equations, have them make guesses for the values of the three variables. Students should discuss
their strategies for making educated rather than random guesses.
6–7 Critique Reasoning, Discussion Groups
Ask students to consider and discuss the following questions:
1. Why are three equations necessary for a word problem that contains three variables?
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2. Would the solution to the problem still be (150, 50, 300) if the constraint from Item 4 were not included as part of the
problem?
Check Your Understanding
Debrief students’ answers to these items to ensure that they understand concepts related to solving a system of equations with
three variables.
Assess
Students’ answers to Lesson Practice problems will provide you with a formative assessment of their understanding of the
lesson concepts and their ability to apply their learning.
See the Activity Practice for additional problems for this lesson. You may assign the problems here or use them as a culmination
for the activity.
Adapt
Check students’ answers to the Lesson Practice to ensure that they understand basic concepts related to writing systems of
equations and solving systems of equations using Gaussian elimination and substitution. If students are having difficulty using
Gaussian elimination, review strategies for reordering the equations and deciding which equation to use to eliminate variables
from the other equations.
Learning Targets
p. 36
Solve systems of three linear equations in three variables using substitution and Gaussian elimination.
Formulate systems of three linear equations in three variables to model a real-world situation.
Close Reading (Learning Strategy)
Definition
Reading text word for word, sentence by sentence, and line by line to make a detailed analysis of meaning
Purpose
Assists in developing a comprehensive understanding of the text
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Vocabulary Organizer (Learning Strategy)
Definition
Using a graphic organizer to keep an ongoing record of vocabulary words with definitions, pictures, notes, and
connections between words
Purpose
Supports a systematic process of learning vocabulary
Note Taking (Learning Strategy)
Definition
Creating a record of information while reading a text or listening to a speaker
Purpose
Helps in organizing ideas and processing information
Summarizing (Learning Strategy)
Definition
Giving a brief statement of the main points in a text
Purpose
Assists with comprehension and provides practice with identifying and restating key information
Paraphrasing (Learning Strategy)
Definition
Restating in your own words the essential information in a text or problem description
Purpose
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Assists with comprehension, recall of information, and problem solving
Graphic Organizer (Learning Strategy)
Definition
Arranging information into maps and charts
Purpose
Builds comprehension and facilitates discussion by representing information in visual form
Group Presentation (Learning Strategy)
Definition
Presenting information as a collaborative group
Purpose
Allows opportunities to present collaborative solutions and to share responsibility for delivering information to an
audience
Think Aloud (Learning Strategy)
Definition
Talking through a difficult text or problem by describing what the text means
Purpose
Helps in comprehending the text, understanding the components of a problem, and thinking about possible paths to a
solution
Identify a Subtask (Learning Strategy)
Definition
Breaking a problem into smaller pieces whose outcomes lead to a solution
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Purpose
Helps to organize the pieces of a complex problem and reach a complete solution
Suggested Learning Strategies
Close Reading, Vocabulary Organizer, Note Taking, Summarizing, Paraphrasing, Graphic Organizer, Group
Presentation, Think Aloud, Identify a Subtask
Sometimes a situation has more than two pieces of information. For these more complex problems, you may need p. 37
to solve equations that contain three variables.
Read and discuss the material on this page with your group before you move on to Example A on the next page.
Use your discussions to clarify the meaning of mathematical concepts and other language used to describe the
information. With your group or your teacher, review background information that will be useful in applying
concepts to the Example.
In Bisbee, Arizona, an old mining town, you can buy souvenir nuggets of gold, silver, and bronze. For $20, you
can buy any of these mixtures of nuggets: 14 gold, 20 silver, and 24 bronze; 20 gold, 15 silver, and 19 bronze; or
30 gold, 5 silver, and 13 bronze. What is the monetary value of each souvenir nugget?
The problem above represents a system of linear equations in three variables. The system can be represented with
these equations.
Although it is possible to solve systems of equations in three variables by graphing, it can be difficult.
Just as the ordered pair (x, y) is a solution of a system in two variables, the ordered triple(x, y, z) is a solution of a
system in three variables. Ordered triples are graphed in three-dimensional coordinate space.
The point (3, −2, 4) is graphed below.
You can use the substitution method to solve systems of equations in three variables.
Example A
p. 38
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Solve this system using substitution.
Step 1:
Solve the first equation for z.
Step 2:
Substitute the expression for z into the second equation. Then solve for y.
Step 3:
Use substitution to solve the third equation for x.
Step 4:
Solve the last equation from Step 2 for y.
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Step 5:
Solve the last equation from Step 1 for z.
Math Tip
As a final step, check your ordered triple solution in one of the original equations to be sure that your
solution is correct.
Solution: The solution of the system is (−4, −6, −3).
Try These A
Solve each system of equations using substitution. Show your work.
a.
(5, −1, 2)
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b.
(3, 0, −4)
Math Terms
When using Gaussian elimination to solve a system of three equations in the variables x, y, and z, you
start by eliminating x from the second and third equations. Then eliminate y from the third equation.
The third equation now has a single variable, z; solve the third equation for z. Then use the value of z
to solve the second equation for y. Finally, use the values of y and z to solve the first equation for x.
Another method of solving a system of three equations in three variables is called Gaussian elimination.
This method has two main parts. The first part involves eliminating variables from the equations in the
system. The second part involves solving for the variables one at a time.
Example B
p. 39
Solve this system using Gaussian elimination.
Step 1:
Use the first equation to eliminate x from the second equation.
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Step 2:
Use the first equation to eliminate x from the third equation.
Step 3:
Use the second equation to eliminate y from the third equation.
Step 4:
Solve the third equation for z.
Step 5:
Solve the second equation for y.
Step 6:
Solve the first equation for x.
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Solution: The solution of the system is (1, 4, 2).
Try These B
a. Solve this system of equations using Gaussian elimination. Show your work.
(−2, 5, 3)
p. 41p. 40
Connect to Math History
The method of Gaussian elimination is named for the German mathematician Carl Friedrich
Gauss (1777–1855), who used a version of it in his calculations. However, the first known use of
Gaussian elimination was a version used in a Chinese work called Nine Chapters of the
Mathematical Art, which was written more than 2000 years ago. It shows how to solve a system
of linear equations involving the volume of grain yielded from sheaves of rice.
1. Work with a partner or with your group. Make a flowchart on notebook paper that summarizes
the steps for solving a system of three equations in three variables by using either
substitution or Gaussian elimination. As you prepare your flowchart to present to the class,
remember to use words and graphics that will help your classmates understand the steps.
Also, be careful to communicate mathematical terms correctly to describe the application of
mathematical concepts.
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Check students’ flowcharts.
A farmer plans to grow corn, soybeans, and wheat on his farm. Let c represent the number of acres
planted with corn, s represent the number of acres planted with soybeans, and w represent the
number of acres planted with wheat.
2. The farmer has 500 acres to plant with corn, soybeans, and wheat. Write an equation in
terms of c, s, and w that models this information.
c + s + w = 500
3. Growing an acre of corn costs $390, an acre of soybeans costs $190, and an acre of wheat
costs $170. The farmer has a budget of $119,000 to spend on growing the crops. Write an
equation in terms of c, s, and w that models this information.
390c + 190s + 170w = 119,000
4. The farmer plans to grow twice as many acres of wheat as acres of corn. Write an equation
in terms of c and w that models this information.
2c = w
5. Write your equations from Items 3–5 as a system of equations.
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Math Tip
Determine the reasonableness of your solution. Does your answer make sense in the context
of the problem?
6. Make sense of problems. Solve the system of equations. Write the solution as an
ordered triple of the form (c, s, w). Explain the meaning of the numbers in the ordered
triple.
(150, 50, 300); The farmer should grow 150 acres of corn, 50 acres of soybeans, and 300 acres of
wheat. This will meet his budget and the requirement to grow twice as many acres of wheat as
corn.
7. Explain what the solution you found in Item 6 represents in the real-world situation.
The farmer should plant 150 acres of corn, 50 acres of soybeans, and 300 acres of wheat.
Check Your Understanding
8. Compare and contrast systems of two linear equations in two variables with systems of
three linear equations in three variables.
Sample answer: For consistent and independent systems, the solution of a system of two linear
equations in two variables is an ordered pair, and the solution of a system of three linear
equations in three variables is an ordered triple. Both types of systems can be solved by using
substitution or a type of elimination.
9. Explain how you could use the first equation in this system to eliminate x from the
second and third equations in the System:
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Sample answer: To eliminate x from the second equation, add the first and second equations in
the system. To eliminate x from the third equation, start by multiplying the first equation by
−2. Then add the resulting equation to the third equation.
Lesson 3-2 Practice
10. Solve the system using substitution.
(−3, 4, 0)
11. Solve the system using Gaussian elimination.
(2, 2, 4)
Use the table for Items 12–14.
Frozen Yogurt Sales
Time Period
Small Cups Sold Medium Cups Large Cups Sold
Sales ($)
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Sold
1:00–2:00
6
10
8
97.60
2:00–3:00
9
12
5
100.80
3:00–4:00
10
12
4
99.20
12. Write a system of equations that can be used to determine s, m, and l, the cost in
dollars of small, medium, and large cups of frozen yogurt.
13. Solve your equation and explain what the solution means in the context of the
situation.
(3.20, 4.00, 4.80); A small cup costs $3.20, a medium cup costs $4.00, and a large cup costs
$4.80.
14. Use appropriate tools strategically. Which method did you use to solve the
system? Explain why you used this method.
Sample answer: I used Gaussian elimination because if I had used substitution instead, I
would have ended up with equations that had fractional coefficients.
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