Lesson 4: Strategies for Solving Simultaneous Equations (Substitution)
Brief Overview of Lesson:
In this lesson students explore an alternate strategy for solving simultaneous linear equations known as substitution. When the solution of a system of linear
equations is an ordered pair that is not located near the origin, or that does not contain whole numbers, the graphic method will not be useful. The substitution
method is just one of many algebraic methods, used to solve a system of equations without using its graphs. As you plan, consider the variability of learners in
your class and make adaptations as necessary.
Prior Knowledge Required:
6.EE.3 Apply the properties of operations to generate equivalent expressions
6.EE.5 Understand solving and equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or
inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true.
6.EE.7 Solve real-world and mathematical problems by writing and solving equations of the form x +p = q and px = q for cases in which p, q, and x are all known
negative rational numbers.
6.EE.9 Use variables to represent two quantities in a real-world problem that change in relationship to one another; write an equation to express one quantity,
thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable.
7.EE.3 Solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and
decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate; and assess the
reasonableness of answers using mental computation and estimation strategies.
7.EE.4 Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by
reasoning about the quantities.
Estimated Time (minutes):
Resources for Lesson:
Two 50 minute lessons
Sample Card Set, Math Teachers’ Café Handout
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Unit: Analyzing and Solving Linear Equations and Pairs of Simultaneous Linear Equations
Content Area/Course: Grade 8 Mathematics
Lesson 4: Strategies for Solving Simultaneous Equations (Substitution)
Time (minutes): Two 50 minute lessons
By the end of this lesson students will know and be able to:
•
Solve simultaneous equations by substitution. (S8)
Essential Question(s) addressed in this lesson:
•
•
How can equations be used to represent real-world and mathematical situations?
Given a particular problem situation, how do we determine which method of solving simultaneous equations will be the most useful?
Standard(s)/Unit Goal(s) to be addressed in this lesson (type each standard/goal exactly as written in the framework):
8.EE.8 Analyze and solve linear equations and pairs of simultaneous linear equations.
a. Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of
intersection satisfy both equations simultaneously.
b. Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection.
For example, 3x + 2y = 5 and 3x + 2y = 6 have no solution because 3x + 2y cannot simultaneously be 5 and 6.
c. Solve real-world and mathematical problems leading to two linear equations in two variables. For example, given coordinates for two pairs of points,
determine whether the line through the first pair of points intersects the line through the second pair.
SMP.7 Look for and make use of structure.
Teacher Content and Pedagogy
There are several methods for solving simultaneous equations, or systems. Students explored graphical models in the previous lesson. When an exact answer is
needed, these graphical models have some limitations. One algebraic method is substitution.
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This lesson is intended to lead students to uncover the limitations of graphical solutions in some situations and to explore other ways to solve systems of
equations.
Do not teach the substitution method as a handy trick or procedure. In fact, it is recommended that the method remain unnamed until the end of the lesson.
Students need to see the connection between this algebraic method and the point of intersection on the graphs of the two lines.
First, students identify if the system has one solution. The point of intersection makes both equations true. If so, students can use the Substitution Property of
Equality to create an equation in one variable.
The Substitution Property of Equality: If a=b, then a can be substituted for b in any expression containing a.
Substitution
Students have previously solved equations using substitution when given an exact value for one of the values. This would be a good reference point from which
to start.
Example 1: Solve the equation y = 5x – 7 when x = -2.
Example 2: Solve the equation y = 5x – 7 when y = 3.
In addition to the Activator Activity, algebra balance scales can also be used to provide a visual model of the concept of substitution.
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Anticipated Student Preconceptions/Misconceptions
Students may not realize that solutions to simultaneous equations are written as a coordinate pair (e.g., x = ___ does not sufficiently describe the point(s) of
intersection.)
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Lesson Sequence
DAY 1
• Opening: Revisit the previous day’s formative assessment: 4x – 2y =3.
Ask students: Were you able to create equations that showed at least one solution, no solutions, or infinite solutions?
Strategically select student work to display on a document reader or on chart paper. Ask students to turn and talk to discuss the strategies that they see.
o What were some strategies for writing the equations?
o What difficulties did they have in finding an equation with one solution, no solution, or infinite solutions?
o What misconceptions do students have?
•
Consider this system: y = 5x and y = 2x + 2. Students graph this system and determine whether there is one solution, no solutions, or many solutions.
Ask students to state the solution. Students should notice the difficulty of finding an exact solution. Using the graphical method, students can only
approximate a solution.
Have students evaluate both equations for the point (2/3, 10/3).
Have students Think, Pair, Share with a partner about the question, “Is there another way to solve this problem other than graphing the system on a
coordinate grid?” “When might it be important to find the exact solution?” Allow for students to talk about their ideas with each other. Have students
share their ideas at the end of this time.
•
If we know that y = 5x and y = 2x + 2, and we know that there is one solution for this system of equations, then there must be one point on the line y =
5x that intersects with one point on the line y = 2x + 2. At that point of intersection, for that same x value, the equations have the same y value. We saw
that when x = 2/3, y was 10/3 in both equations. So, (2/3, 10/3) is the point of intersection. Does that mean that 5x and 2x + 2 have the same value at that
point?
Y = 5x
y = 2x + 2
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If y = y at that point of intersection, is it true that 5x = 2x + 2?
Can you solve this equation? (Algebraic approach to solving systems.) Students will find the x value. They have to realize that in a system of equations,
the solution is a point, with both an x and y value.
•
Post the next two problems on the board and have students try them on their own. Students pair up. One person solves it graphically and finds the point of
intersection. The other person solves it algebraically. Compare solutions with each other, and then with another pair.
Problem 1:
y=x–2
y = -x + 12
(Solution: (7,5))
Problem 2:
y = -3x
x + y = -2
(Solution: (1, -3))
The word substitute or substitution may come up in the discussions. Mention that this method is often called the substitution method. Ask students
why.
Connect algebraic representations to visual representations. Problems can be described and solved in multiple ways.
As a Ticket to Leave, ask: When might you use the graphing method, and when might you use the substitution method?
Students need to analyze the structure of the given equations in order to determine which method to use (SMP.7)
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DAY 2
•
Equation Cards Activity
1. Students work in pairs, using the Sample Cards to practice solving simultaneous equations algebraically. Each pair is given a card set in zip-lock bags. Does
the pair of equations have one, no, or infinite solutions?
Each zip-lock bag contains two cards. Cards are folded so that there is a front and a back to each card. Cards #1-5 have y isolated in both equations so
students can set the equations equal to each other. Cards # 6 – 10 have either x or y isolated so that one equation can be substituted into the other.
2. Students will then solve the simultaneous equations algebraically using substitution. They put their cards back in their zip-lock bag, and swap with another
team. Repeat.
3. Have a few groups go to the board, post, and explain their solutions.
•
Students revisit two examples from the previous day:
Example 1: y = x – 2
y = -x + 12
•
Example 2: y = 3x
x + y = 20
They graph each pair of equations. How do the solutions compare to those derived from using substitution method on the previous day?
Closing: Ticket to Leave: Students write a response: What are the advantages of solving simultaneous equations with graphing? What are the advantages
of solving them with the substitution method? Which one makes more sense to you?
Assign Homework: Math Teachers’ Café (practice solving simultaneous equations with substitution)
Preview outcomes for the next lesson: Students will be able to use a variety of strategies to solve systems of equations, including graphing, substitution, and
elimination.
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Lesson 4: Sample Card Set
Card Set #1:
y
5x–8
y
5x+3
y
7 x – 21
Card Set #2:
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y
5x+9
y
-3 x + ¼
Card Set #3
y
-5 x – ¾
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Card Set #4
y
2/3 x + 7
y
1/3 x + 9
y
-2 x + 7
y
–x –x + 3 + 4
Card Set #5
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Card Set #6
4x–1
y
2x -y=5
2x – (
)=5
Card Set #7
2y+4
x
2x – y = 5
2(
)–y =5
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Card Set #8
2/5 x – 3
y
-2x = -3 – y
5
-2x = –3 – (
5
)
Card Set #9
-1/4 y + 3
x
-4x + y = 20
-4 (
) + y = 20
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Card Set #10
1/3x – 7
y
–x + y = 1
3
2
–x + (
3
)= 1
2
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Answer Key
Card Set #1
No Solution -8 ≠ 3
Card Set #2
One Solution (15, 84)
Card Set #3
One Solution (-1/2, 7/4)
Card Set #4
One Solution (6, 11)
Card Set #5
Infinitely Many Solutions -2x + 7 = -x –x +3 + 4
Card Set #6
One Solution (-2, -9)
Card Set #7
One Solution (-1, 2)
Card Set #8
Infinitely Many Solution 2/5x – 3 = 2/5x – 3
Card Set #9
One Solution (7/8, 17/2)
Card Set #10
No Solution -7 ≠ ½
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Lesson 4: Math Teachers’ Café
Name: _________________________________ ____ Class: __________ Date:___________________
Solving Systems Using Substitution – Jokingly
Directions – Show all work on a separate piece of paper. Solve each system of equations by the substitution
method. Then, write the word next to the correct answer in the box containing the selected solutions.
What’s the best dessert in the Math Teachers’ Café?
Answer to #4
Answer to #1
Answer to #6
Answer to #7
Answer to #9
!
1. y = 5x
x + y = 30
2. 2x + 5y = 26
y = x/4
4. 4p + 2q = 10
p – 5q = 8
5. 5c – 2d = 0
6d = 1 + c
3. B – 3a = 1
b = 4a
6. 3s – t = 1
3s = 9
7. The sum of two numbers is 36 and their difference is 6. What are the numbers?
8. There are 312 students at the school dance. There are 48 more girls than boys. How many girls
are at the dance? How many boys are at the dance?
9. The school’s Drama Club charges $10 for an adult ticket and $5 for a child’s ticket. One night 410
tickets are sold for a total of $3625. How many adults attended the show? How many children
attended the show?
Answer Options:
( 180, 132)
(3, -1)
(5, 25)
vanilla
A
slice
(3,8)
(1, 4)
( 315, 95)
of
piece
pi
( 24, 6 )
One
( 21, 15)
chocolate
(1/14, 5/28)
cake
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Math Teachers’ Café ANSWER KEY
Solving Systems Using Substitution – Jokingly
Directions – Show all work on a separate piece of paper. Solve each system of equations by the substitution
method. Then, write the word next to the correct answer in the box containing the selected solutions.
What’s the best dessert in the Math Teachers’ Café?
Answer to #4
A
9. y = 5x
x + y = 30
(5, 25)
4. 4p + 2q = 10
p – 5q = 8
(3, -1)
Answer to #1
slice
Answer to #6
Answer to #7
Answer to #9
of
chocolate
pi !
2. 2x + 5y = 26
y = x/4
3. B – 3a = 1
b = 4a
(24, 6)
(1, 4)
5. 5c – 2d = 0
6d = 1 + c
(1/14, 5/28)
6. 3s – t = 1
3s = 9
(3, 8)
7. The sum of two numbers is 36 and their difference is 6. What are the numbers?
(21, 15)
8. There are 312 students at the school dance. There are 48 more girls than boys. How many girls
are at the dance? How many boys are at the dance?
(180, 132)
10. The school’s Drama Club charges $10 for an adult ticket and $5 for a child’s ticket. One night 410
tickets are sold for a total of $3625. How many adults attended the show? How many children
attended the show?
(315, 95)
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