Lesson 7
Reasoning about Solving Equations (Part 1)
Problem 1
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.
![A coordinate plane, origin O labeled. Horizontal axis, mass in grams, 0 to 22 by twos. Veritcal axis, volume in liters, 0 to 180 by twenties, a point labeled, 5 comma 28.](https://cms-im.s3.amazonaws.com/cWumQCcLqW7WBYzSEJdZEo11?response-content-disposition=inline%3B%20filename%3D%227-7.2.E.PP.Image.08.png%22%3B%20filename%2A%3DUTF-8%27%277-7.2.E.PP.Image.08.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240630%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240630T181915Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=0e055e2a296e335a7f309637e30476b4ac22cf32bddd4eec1d16ecb5fade6296)
- What is the constant of proportionality in this relationship?
- In this situation, what is the meaning of the number you found in part a?
- Add at least three more points to the graph above, and label with their coordinates.
- 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.
Solution
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(From Unit 2, Lesson 11.)Problem 2
Explain how the parts of the balanced hanger compare to the parts of the equation.
\(7=2x+3\)
![Balanced hanger, left side, 7 blue squares, right side, 2 red circles, 3 blue squares.](https://cms-im.s3.amazonaws.com/inMChNH4d532hq7vWJYqCSYf?response-content-disposition=inline%3B%20filename%3D%227-7.6.B1.newPP.01.png%22%3B%20filename%2A%3DUTF-8%27%277-7.6.B1.newPP.01.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240630%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240630T181915Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=2ae2410cba8329c3fe1f4c6cb276328680ae8e5341faa8251393d5d83a07dc1a)
Solution
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Problem 3
For the hanger below:
- Write an equation to represent the hanger.
- Draw more hangers to show each step you would take to find \(x\). Explain your reasoning.
- Write an equation to describe each hanger you drew. Describe how each equation matches its hanger.
![Balanced hanger, left side, 5 circles labeled x, 1 square labeled 2, right side, rectangle labeled 17.](https://cms-im.s3.amazonaws.com/JLQRDzvTPpAjLUa1j7Jo33pS?response-content-disposition=inline%3B%20filename%3D%227-7.6.B1.newPP.04.png%22%3B%20filename%2A%3DUTF-8%27%277-7.6.B1.newPP.04.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240630%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240630T181915Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=c959d33c8d6b41adf2fec16947e6797566c3251c41cc19026affe5b4768a10d5)
Solution
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