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.

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.
 

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.

Some students may not remember to find the value of the second variable after finding the first. They may need a reminder that the solution to a system of linear equations is a pair of values.

If some students struggle with the last system because the variable that is already isolated is equal to an expression rather than a number, ask what they would do if the first equation were  instead of .

If students don't know how to approach the last system, ask them to analyze both equations and see if the value of one of the variables could be found easily.

Select previously identified students to share their responses and strategies. Display their work for all to see. Highlight the strategies that involve substitution, and name them as such.

Make sure students see that the last two equations can be solved by substituting in different ways. Here are two ways for solving the third system, , by substitution:

Here are two ways of solving the last system, , by substitution:

In each of these two systems, students are likely to notice that one way of substituting is much quicker than the other. Emphasize that when one of the variables is already isolated or can be easily isolated, substituting the value of that variable (or the expression that is equal to that variable) into the other equation in the system can be an efficient way to solve the system.

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.

When solving the second system, students are likely to substitute the expression for in the first equation, . Done correctly, it should be written as . Some students may neglect to write parentheses and write . Remind students that if is equal to , then  is 2 times or . (Alternatively, use an example with a sum of two numbers for : Suppose , which means , or 20. If we express as a sum of 3 and 7, or , then , not . The latter has a value of 13, not 20.)

Some students who correctly write may fail to distribute the subtraction and write the left side as . Remind them that subtracting by can be thought of as adding and ask how they would expand this expression.

Select previously identified students to share their responses and reasoning. Display their work for all to see.

Highlight the different ways to perform substitutions to solve the same system. For example:

• In the first system, can be substituted for , or could be substituted for .
• In the fourth system, , we could substitute for  in the second equation, or we could substitute for  in the second equation.