6.2.1: Reasoning about Solving Equations (Part 1)

Lesson

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

Exercise \(\PageIndex{1}\): 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

Figure \(\PageIndex{1}\): Two hanger diagrams. First hanger, unbalanced with left side lower, left side, green triangle, right side, blue square. Second hanger, balanced, left side, green triangle, right side, three blue squares, one red circle.

Exercise \(\PageIndex{2}\): 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.
   • \(2 ?+3=5\)
   • \(3 ?+2=3\)
   • \(6=2?+3\)
   • \(7=3?+1\)
2. Find the solution to each equation. Use the hanger to explain what the solution means.

Figure \(\PageIndex{2}\): Four balanced hanger diagrams, A, B, C, D. Diagram A, left side, 7 squares, 1, right side, three circles, w, 1 square, 1. Diagram B, left side, 2 triangles, z, 3 squares, 1, right side, 5 squares, 1. Diagram C, left side, 3 pentagons, x, 2 squares, 1, right side, 3 squares, 1. Diagram D, left side, 6 squares, right side, 1, 2 crowns, y, 3 squares, 1.

Exercise \(\PageIndex{3}\): 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.

Figure \(\PageIndex{3}\): Four balanced hanger diagrams, A, B, C, D. Diagram A, left side, rectangle, 7, right side, 3 circles, x, square, 1. Diagram B, left side, 2 diamonds, y, square, 10, right side, rectangle, 31. Diagram C, left side, rectangle 6.8, right side 2 pentagons, z, square 2.2. Diagram D, left side, 4 triangles, w, square, 3 over 2, right side, rectangle, 17 over 2.

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 \(7=3x+1\).

Figure \(\PageIndex{4}\)

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.

Figure \(\PageIndex{5}\): Balanced hanger, left side, 6 blue squares and one red squared being removed. Right side, 3 green squares, and one red square being removed. To the side, an equation 7 minus 1 = 3 + 1 minus 1, with each minus 1 written in red.

An equation for the new balanced hanger is \(6=3x\).

Figure \(\PageIndex{6}\)

So the hanger will balance with \(\frac{1}{3}\) of the weight on each side: \(\frac{1}{3}\cdot 6=\frac{1}{3}\cdot 3x\).

Figure \(\PageIndex{7}\): Balanced hanger, left side 6 blue squares, right side, three green circles. A dotted line is drawn around each of 3 groups, each groups consists of two blue square s from the left side and one green circle from the right side. To the side, an equation says 6=3x.

The two sides of the hanger balance with these weights: 6 1-unit weights on one side and 3 weights of unknown size on the other side.

Figure \(\PageIndex{8}\)

Here is a concise way to write the steps above:

\(\begin{array}{lr}{7=3x+1}&{}\\{6=3x}&{\text{after subtracting 1 from each side}}\\{2=x}&{\text{after multiplying each side by }\frac{1}{3}}\end{array}\)

Practice

Exercise \(\PageIndex{4}\)

There is a proportional relationship between the volume of a sample of helium in liters and the mass of that sample in grams. If the mass of a sample is 5 grams, its volume is 28 liters. (5, 28) is shown on the graph below.

Figure \(\PageIndex{9}\)

1. What is the constant of proportionality in this relationship?
2. In this situation, what is the meaning of the number you found in part a?
3. Add at least three more points to the graph above, and label with their coordinates.
4. Write an equation that shows the relationship between the mass of a sample of helium and its volume. Use \(m\) for mass and \(v\) for volume.

(From Unit 2.4.2)

Exercise \(\PageIndex{5}\)

Explain how the parts of the balanced hanger compare to the parts of the equation.

\(7=2x+3\)

Figure \(\PageIndex{10}\)

Exercise \(\PageIndex{6}\)

For the hanger below:

1. Write an equation to represent the hanger.
2. Draw more hangers to show each step you would take to find \(x\). Explain your reasoning.
3. Write an equation to describe each hanger you drew. Describe how each equation matches its hanger.

Figure \(\PageIndex{11}\)