This is a math puzzle in which the goal is to place the given equations into a table where all the rows contain equivalent equations. Each row represents a group of equations that are algebraically the same but are expressed in different forms.

Given Equations on the Cards:

1. $5y - 6 = 2x$
2. $5y = 6 - 2x$
3. $2x - 6 = 5y$
4. $5y - 2x = 6$

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Steps to Solve:

The equations need to be rearranged so they match the form of the equations already in the table. To do this:

• Rearrange each card to isolate terms and match other rows' structure.
• Verify they are equivalent using algebra.

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Matching the Cards to Rows:

1. Card $5y - 6 = 2x$: Rearrange:
   $5y = 6 + 2x$ → Matches the second row, second column.

2. Card $5y = 6 - 2x$: This is already simplified and matches the form of an equivalent equation.
   Rearrange: $5y - 6 = -2x$ → Matches row one, left column.

3. Card $2x - 6 = 5y$: Rearrange:
   $2x = 6 + 5y$ → Matches top row, first column.

4. Card $5y - 2x = 6$: Rearrange:
   $5y + 2x = 6$ → Matches the third row, first column.

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Filled Table Solution:

First Row	Second Row	Third Row
$2x = 6 + 5y$	$2x - 6 = 5y$	$5y + 2x - 6 = 0$
$5y - 6 = 2x$	$5y = 6 + 2x$	$5y + 2x = 6$

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Want more details or help with algebraic rearrangement? Let me know!

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5 Related Questions:

1. How can you check if two equations are equivalent?
2. What are the rules for rearranging algebraic equations?
3. How do you write an equation in standard form?
4. What strategies can help group equivalent equations?
5. How can you graph these equations to verify their equivalence?

Tip: When solving for equivalence, isolate one variable (like $x$ or $y$) in all equations and compare the forms.