# Lesson 7

Reasoning about Solving Equations (Part 1)

Let’s see how a balanced hanger is like an equation and how moving its weights is like solving the equation.

### 7.1: Hanger Diagrams

In the two diagrams, all the triangles weigh the same and all the squares weigh the same.

For each diagram, come up with . . .

1. One thing that must be true
2. One thing that could be true
3. One thing that cannot possibly be true

### 7.2: Hanger and Equation Matching

On each balanced hanger, figures with the same letter have the same weight.

1. Match each hanger to an equation. Complete the equation by writing $$x$$, $$y$$, $$z$$, or $$w$$ in the empty box.

• $$2 \boxed{\phantom{3}} + 3 = 5$$
• $$3 \boxed{\phantom{3}} + 2 = 3$$
• $$6 = 2 \boxed{\phantom{3}} + 3$$
• $$7 = 3 \boxed{\phantom{3}} + 1$$

2. Find the solution to each equation. Use the hanger to explain what the solution means.

### 7.3: Use Hangers to Understand Equation Solving

Here are some balanced hangers where each piece is labeled with its weight. For each diagram:

1. Write an equation.
2. Explain how to figure out the weight of a piece labeled with a letter by reasoning about the diagram.
3. Explain how to figure out the weight of a piece labeled with a letter by reasoning about the equation.

### Summary

In this lesson, we worked with two ways to show that two amounts are equal: a balanced hanger and an equation. We can use a balanced hanger to think about steps to finding an unknown amount in an associated equation.

The hanger shows a total weight of 7 units on one side that is balanced with 3 equal, unknown weights and a 1-unit weight on the other. An equation that represents the relationship is $$7=3x+1$$.

We can remove a weight of 1 unit from each side and the hanger will stay balanced. This is the same as subtracting 1 from each side of the equation.

An equation for the new balanced hanger is $$6=3x$$.

So the hanger will balance with $$\frac13$$ of the weight on each side: $$\frac13 \boldcdot 6 = \frac13 \boldcdot 3x$$.
\begin {align} 7&=3x+1 & \\ 6&=3x & \text{after subtracting 1 from each side} \\ 2 &= x & \text{after multiplying each side by } \tfrac13 \\ \end{align}