Lesson 12
Balanced Moves
Let's rewrite equations while keeping the same solutions.
Problem 1
In this hanger, the weight of the triangle is \(x\) and the weight of the square is \(y\).
![Balanced hanger. Left side, 1 triangle, 3 squares. Right side, 4 triangles, 1 square.](https://cms-im.s3.amazonaws.com/27BuaEB4nFGe4nBhBRKpJUgd?response-content-disposition=inline%3B%20filename%3D%228-8.4.PP.B.Image.03.png%22%3B%20filename%2A%3DUTF-8%27%278-8.4.PP.B.Image.03.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=20240630T182042Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=441698be8e6af593e57f048d7dfdcd5488625693b62c0b836b072acb2fa7e2b6)
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Write an equation using \(x\) and \(y\) to represent the hanger.
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If \(x\) is 6, what is \(y\)?
Problem 2
Andre and Diego were each trying to solve \(2x+6=3x-8\). Describe the first step they each make to the equation.
- The result of Andre’s first step was \(\text-x+6=\text-8\).
- The result of Diego’s first step was \(6=x-8\).
Problem 3
Match each set of equations with the move that turned the first equation into the second.
Problem 4
What is the weight of a square if a triangle weighs 4 grams?
Explain your reasoning.
![Balanced hanger. Left side, 1 triangle, 2 squares. Right side, 3 triangles, 1 square.](https://cms-im.s3.amazonaws.com/5zwMaXudnndqJMHzdDAD5caD?response-content-disposition=inline%3B%20filename%3D%228-8.4.PP.B.Image.01.png%22%3B%20filename%2A%3DUTF-8%27%278-8.4.PP.B.Image.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=20240630T182042Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=41f2767523bfc79d3e9758788ae5259f9ecc7a103b3c15ba4999a5615c2caf1f)
Problem 5
Here is a balanced hanger diagram.
Each triangle weighs 2.5 pounds, each circle weighs 3 pounds, and \(x\) represents the weight of each square. Select all equations that represent the hanger.
![A balanced hanger. Left side, 4 squares, 2 triangles, 2 circles. Right side, 2 squares, 1 triangle, 3 circles.](https://cms-im.s3.amazonaws.com/uqP5ncBED5WrQCNgFW2DgG5z?response-content-disposition=inline%3B%20filename%3D%228-8.4.B2.PP.hang7.png%22%3B%20filename%2A%3DUTF-8%27%278-8.4.B2.PP.hang7.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=20240630T182042Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=283b966eb945a17d7664522e819e8a6e3e04e7aaff141778d01f6f033da01219)
\(x+x+x+x+11=x+11.5\)
\(2x=0.5\)
\(4x+5+6=2x+2.5+6\)
\(2x+2.5=3\)
\(4x+2.5+2.5+3+3=2x+2.5+3+3+3\)