## Using Inverse Operations to Solve Equations: Overview

Earlier, your students reviewed the terms expression, algebraic expression, and equation. They wrote and then simplified or evaluated expressions or solved equations involving addition and subtraction. (See Expressions and Equations.) Now, following a similar sequence, students will work with expressions and equations involving multiplication and division.

In this chapter, students begin by using algebraic expressions to describe patterns. They will write and evaluate expressions to solve problems on hourly or weekly wages or on the cost of buying several of the same item. For example, 9n is an expression showing the amount a person would earn by dog-sitting for n number of days at \$9.00 per day. Although students have written and evaluated expressions before, you may need to remind them that any letter can be used as a variable. Explain that they can substitute any number for the variable. Unlike a single-variable equation, in which the variable has only one value, an expression allows you to substitute different values for the variable to find different results. So in the above example, you could find the amount a person would earn by dog-sitting for 2, 6 or 20 days by substituting 2, 6, or 20 for the variable.

Students will also write expressions involving two operations. For example, if a person charges \$9.00 per day to dog-sit, plus a \$10 grooming fee, you would use the expression 9n + 10 to find the amount earned dog-sitting for n number of days. When evaluating two-step expressions, students may need to follow the order of operations. To evaluate, start with any operation within parentheses. Then multiply or divide in order from left to right. Then add or subtract in order from left to right.

 Evaluate when n = 3. 12 ÷ (3 + n) = 12 ÷ (3 + 3) 12 ÷ 6 = 2
 Evaluate when q = 4. 11 − 2q = 11 − (2 x 4) 11 − 8 = 3

In the first lesson, students focus on writing word phrases as algebraic expressions and writing algebraic expressions in words. These skills are the foundation for writing equations correctly. At this point you will want to review the Commutative Property. The Commutative Property states that when you multiply or add numbers, changing the order of the numbers will not change the result. So, 6 + 3 = 3 + 6 and 3 x 6 = 6 x 3

In accordance with this property, expressions involving addition and multiplication can be correctly written several ways. For example, 6 times a number can be written as 6n, n x 6, 6 x n, n · 6 and 6 · n. Similarly, a number plus three can be written as either n + 3 or 3 + n. To reinforce the validity of the Commutative Property, write the same multiplication or addition expression several different ways. Then have students evaluate it to see that the results are the same. Have students experiment with the Commutative Property with expressions involving subtraction and division to reinforce that the property does not apply to these operations.

Once students have practiced writing and evaluating algebraic expressions, they will solve equations that involve multiplication and division. Students are taught two different ways to solve equations. One method is to make a function table. A function is a rule that gives exactly one value of y for each value of x. For the function y = 2x, the value of y is always 2 times the value of x.

Students can complete a function table to find the value for r that when multiplied by 3 yields the product 21. In the function table, they list possible values for r in the left-hand column, substitute them in the equation, and list the outcomes in the right-hand column until they reach the desired outcome of 21. To find the value for r that when divided by 3 gives the quotient 3, they follow the same procedure until they reach the desired outcome of 3.

3r = 21
Value for r Outcome
1 3
2 6
3 9
4 12
5 15
6 18
7 21

r = 7
r ÷ 3 = 3
Value for r Outcome
3 1
4 1 R1
5 1 R2
6 2
7 2 R1
8 2 R2
9 3

r = 9

For more on functions and function tables see Teacher Support, Grade 5, Functions.

Inverse operations can also be used to solve equations. Inverse operations are operations that undo each other. If you multiply two numbers and then divide by one of them, you undo the original multiplication. Thus, multiplication and division are inverse operations. For example, if you know that a number multiplied by 3 is 18, then 18 ÷ 3 must be equal to that number. Addition and subtraction are also inverse operations. Students have worked with inverse operations before. They have used inverse operations to solve addition and subtraction equations; they have added to check their work in subtraction problems; they have multiplied to check their work in division problems. They have also used inverse operations when writing and working with fact families and related facts.

To solve an equation in which a variable is multiplied by a number, students will learn to divide both sides of the equation by that number. So, to solve 3r = 21, students divide both sides by three.

 3r = 21 (3r) ÷ 3 = 21 ÷ 3 r = 7

To solve an equation in which a variable is divided by a number, students will learn to multiply both sides of the equation by that number. So, to solve r ÷ 3 = 3, students will multiply both sides by 3.

 r ÷ 3 = 3 3(r ÷ 3) = 3 x 3 r = 9

At first, students will use inverse operations without understanding the concept of performing the same operation on both sides of the equation. Initially, what they will think of as solving by inverse operations is really just solving by use of a related fact. For example, a student looking at 6t = 42 may think of the fact 42 ÷ 6 = 7 and use it to solve. After introducing the concept of inverse operations, your goal is to get students to understand to balance an equation, they are actually performing the same operation on both sides of the equation. So, in the above example, they are "removing" the 6 from both sides of the equation by dividing both sides of the equation by 6. This is an important concept that students will need to understand when they solve more complex algebraic equations.

Students may need to model equations to understand that when the same operation is performed on both sides of an equation the variable is not changed. Begin by using counters to model and solve a simple equation.

Then have them add, subtract, multiply or divide each side of the equation by the same number. To show that the value of the variable remains the same, have them substitute the value of the variable from the original equation into the new one.

This provides a good opportunity to review the Distributive Property. Applying this property can make it easier to solve an addition or subtraction equation when you are multiplying both sides by a number. The Distributive Property states that when you multiply two addends by a factor, the answer is the same as if you multiply each addend by the factor and then add the products.

 6(5 + 3) = (6 x 5) + (6 x 3) 6 x 8 = 30 + 18 48 = 48

The Distributive Property is also true for differences.

 6(5 − 3) = (6 x 5) − (6 x 3) 6 x 2 = 30 − 18 12 = 12

Once students understand that using inverse operations to solve an equation requires performing the same operation on both sides of the equation, have them apply this knowledge by writing equations to solve word problems. As students practice writing equations, make sure that they “translate” the words in the problem into the correct operation or operations. Also, make sure that they are using inverse operations by performing the same operation on both sides of the equation.

For example:

Jill ran the same number of laps each day for a week. At the end of the week, she had run 42 laps. How many laps did she run each day? (7d = 42; d = 6)