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Unit 4, Lesson 13

Solving Systems by Substitution

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Integrated Math 1

13.1

Warm-up

Standards Alignment

Building On

Addressing

Building Toward

Instructional Routines

None

Materials

None

Activity Narrative

This activity uses pictures to set up the idea of solving systems by substitution. Students may be familiar with this type of problem in which a small detail change is important to solving the system. No such trick is given in this problem, but the idea can be addressed if students bring it up.

While several valid strategies exist, focus attention on solving for the value of each picture and using that value to find additional values for other pictures. This primes student thinking for using substitution to solve systems of linear equations.

Launch

Arrange students in groups of 2. Display the image for all to see. Give students 1 minute of quiet think time, then ask students if they have ever seen a similar type of problem outside of class. If it is unclear, tell students that they should try to find the numerical value of the last expression by using the information in the other equations. Give students 5–7 minutes to solve the problem with their partner, then follow with a whole-class discussion.

Activity

None

Find the value of the last line. Be prepared to explain your reasoning.

system of equations using stars, suns, and rockets

Student Response

to access Student Response.

Building on Student Thinking

Activity Synthesis

Ask a group to share their strategy to get started, and stop them when they offer the value of one of the symbols. Ask another group how they used that information to get the value of another symbol. Continue selecting groups until the value of each symbol is known and the value of the last line is shared.

If it does not come up in the discussion of the strategies, point out that the first three equations could be written as a system.

The solution to this system is and can be found by solving each equation from the first to the third, using known information.

13.2

Activity

Instructional Routines

None

Materials

None

Activity Narrative

In this activity, students extend the reasoning they used in the Warm-up to solve systems of two equations. Some students may choose to solve by graphing, but the systems lend themselves to being solved efficiently and precisely by substitution.

As students work, pay attention to the methods that students use to solve the systems. Identify those who solve by substitution—by replacing a variable or an expression in one equation with an equal value or equivalent expression from the other equation. Ask these students to share later.

Launch

Arrange students in groups of 2. Ask students to recall what it means for a pair of values to be a solution to a system. Give students 6–8 minutes of quiet time to solve as many systems as they can and then a couple of minutes to share their responses and strategies with their partner.

Engagement: Internalize Self Regulation. Chunk this task in half to differentiate the degree of difficulty or complexity. Present and discuss the first two problems before showing the last two. Support students in metacognitive development by having students articulate the steps that they used when they solved by substitution in these more straightforward examples. Record the steps on a chart or whiteboard to encode the process for later retrieval. Then, reveal the remaining problems.
Supports accessibility for: Organization, Attention

13.3

Activity

Instructional Routines

MLR7: Compare and Connect

Materials

None

Activity Narrative

This activity allows students to practice solving systems of linear equations by substitution and reinforces the idea that there are multiple ways to perform substitution. Students are directed to find the solutions without graphing.

Monitor for the different ways that students use substitutions to solve the systems. Invite students with different approaches to share later.

Launch

Keep students in groups of 2. Give students a few minutes to work quietly and then time to discuss their work with a partner. If time is limited, ask each partner to choose two different systems to solve. Follow with a whole-class discussion.

Activity

None

Lesson Synthesis

To emphasize that the method we choose for solving a system may depend on the system, and that some systems are more conducive to being solved by substitution than others are, present the following systems to students:

System 1

System 2

System 3

Ask students to choose a system and make a case (in writing, if possible) for why they would or would not choose to solve that system by substitution. Consider asking students to use sentence starters such as these:

"I would choose to solve system by substitution because . . . ."
"I would not choose to solve system by substitution because . . . ."

With a little bit of rearrangement, all systems could be solved by substitution without cumbersome computation, but System 2 would be most conducive to solving by substitution.

Consider collecting students' responses or asking them to share their written arguments with a partner.

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Student Lesson Summary

The solution to a system can usually be found by graphing, but graphing may not always be the most precise or the most efficient way to solve a system.

Here is a system of equations:

The graphs of the equations show an intersection at approximately 20 for and approximately 10 for .

Without technology, however, it is not easy to tell what the exact values are.

<p>Coordinate plane, origin O. Horizontal axis from 0 to 30 by 5’s, labeled p. Vertical axis from 0 to 20 by 5’s, labeled q. Two lines intersect near 20 comma 10.</p>

Instead of solving by graphing, we can solve the system algebraically. Here is one way.

If we subtract from each side of the first equation, , we get an equivalent equation: . Rewriting the original equation this way allows us to isolate the variable .

Because is equal to , we can substitute the expression in the place of in the second equation. Doing this gives us an equation with only one variable, , and makes it possible to find .

Now that we know the value of , we can find the value of by substituting 20.2 for in either of the original equations and solving the equation.

The solution to the system is the pair and , or the point on the graph.

This method of solving a system of equations is called solving by substitution, because we substituted an expression for into the second equation.

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Lesson 13

Solving Systems by Substitution

Warm-up

Standards Alignment

Building On

8.EE.C

Addressing

Building Toward

HSA-REI.C.6

Instructional Routines

Materials

Activity Narrative

Launch

Activity

Student Response

Building on Student Thinking

Activity Synthesis

Activity

Instructional Routines

Materials

Activity Narrative

Launch

Activity

Instructional Routines

Materials

Activity Narrative

Launch

Activity

Lesson Synthesis

Student Lesson Summary

Activity

Student Response

Building on Student Thinking

Activity Synthesis

Student Response

Building on Student Thinking

Activity Synthesis

Standards Alignment

Building On

Addressing

HSA-REI.C.6

Building Toward

Standards Alignment

Building On

Addressing

HSA-REI.C.6

Building Toward