Preview of the cut-and-paste activity where students solve equations to categorize them by solution type.

It looks like you've uploaded an image of a "One, No, and Infinite Solutions Card Sort" activity. This is a common classroom task used to help students classify linear equations based on the number of solutions they have: one solution, no solution, or infinitely many solutions.

Let’s go through each equation, solve it, and determine how many solutions it has. Then I’ll explain the reasoning.

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🔹 Step-by-step Analysis

We'll evaluate each equation and classify it:

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1. $ b - 3 - 6b = 2 $
Simplify:
$ -5b - 3 = 2 $
$ -5b = 5 $
$ b = -1 $
✔ One Solution

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2. $ -3(-4 - k) = 3(k + 4) $
Left: $ -3(-4 - k) = 12 + 3k $
Right: $ 3(k + 4) = 3k + 12 $
So: $ 12 + 3k = 3k + 12 $
Subtract $ 3k $ from both sides: $ 12 = 12 $
✔ Infinite Solutions (always true)

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3. $ -5p + 5p = 0 $
$ 0 = 0 $
✔ Infinite Solutions (true for all p)

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4. $ 7x - 1 = -(1 + 3x) + 10x $
Right: $ -1 - 3x + 10x = -1 + 7x $
So: $ 7x - 1 = 7x - 1 $
Subtract $ 7x $: $ -1 = -1 $
✔ Infinite Solutions

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5. $ 50 + 9u = 4(1 + 8u) $
Right: $ 4 + 32u $
So: $ 50 + 9u = 4 + 32u $
$ 50 - 4 = 32u - 9u $ → $ 46 = 23u $ → $ u = 2 $
✔ One Solution

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6. $ -3 + 9n = -5(1 - 2n) - n $
Right: $ -5 + 10n - n = -5 + 9n $
So: $ -3 + 9n = -5 + 9n $
Subtract $ 9n $: $ -3 = -5 $ ✘ No Solution

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7. $ -12 - 6k = 3(-4 - 2k) $
Right: $ -12 - 6k $
So: $ -12 - 6k = -12 - 6k $
✔ Infinite Solutions

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8. $ 103 = 7(x + 1) + 5x $
Right: $ 7x + 7 + 5x = 12x + 7 $
So: $ 103 = 12x + 7 $
$ 96 = 12x $ → $ x = 8 $
✔ One Solution

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9. $ 9x + 1 = -9(-1 - x) $
Right: $ -9(-1 - x) = 9 + 9x $
So: $ 9x + 1 = 9x + 9 $
Subtract $ 9x $: $ 1 = 9 $ ✘ No Solution

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10. $ -3n + 3n = 5 $
$ 0 = 5 $ ✘ No Solution

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11. $ -7x + 2 = 2(3 + 2x) - 11x $
Right: $ 6 + 4x - 11x = 6 - 7x $
So: $ -7x + 2 = -7x + 6 $
Add $ 7x $: $ 2 = 6 $ ✘ No Solution

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12. $ -8(p + 7) = -88 $
Left: $ -8p - 56 = -88 $
$ -8p = -32 $ → $ p = 4 $
✔ One Solution

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13. $ 5n = 5n + 2 $
Subtract $ 5n $: $ 0 = 2 $ ✘ No Solution

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14. $ 20 + 2x = 2(10 - 3x) + 8x $
Right: $ 20 - 6x + 8x = 20 + 2x $
So: $ 20 + 2x = 20 + 2x $
✔ Infinite Solutions

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🔹 Final Classification Summary

#### ✔ One Solution
- $ b - 3 - 6b = 2 $
- $ 50 + 9u = 4(1 + 8u) $
- $ 103 = 7(x + 1) + 5x $
- $ -8(p + 7) = -88 $

#### ✘ No Solution
- $ -3 + 9n = -5(1 - 2n) - n $
- $ 9x + 1 = -9(-1 - x) $
- $ -3n + 3n = 5 $
- $ -7x + 2 = 2(3 + 2x) - 11x $
- $ 5n = 5n + 2 $

#### 🔁 Infinite Solutions
- $ -3(-4 - k) = 3(k + 4) $
- $ -5p + 5p = 0 $
- $ 7x - 1 = -(1 + 3x) + 10x $
- $ -12 - 6k = 3(-4 - 2k) $
- $ 20 + 2x = 2(10 - 3x) + 8x $

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🔍 How to Determine the Number of Solutions

- One Solution: The equation simplifies to something like $ x = 5 $. Only one value satisfies it.
- No Solution: Simplifies to a false statement like $ 0 = 5 $. Never true.
- Infinite Solutions: Simplifies to a true statement like $ 0 = 0 $. True for all values.

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✔ Answer Key for Card Sort

| Equation | Solution Type |
|--------|---------------|
| $ b - 3 - 6b = 2 $ | One Solution |
| $ -3(-4 - k) = 3(k + 4) $ | Infinite Solutions |
| $ -5p + 5p = 0 $ | Infinite Solutions |
| $ 7x - 1 = -(1 + 3x) + 10x $ | Infinite Solutions |
| $ 50 + 9u = 4(1 + 8u) $ | One Solution |
| $ -3 + 9n = -5(1 - 2n) - n $ | No Solution |
| $ -12 - 6k = 3(-4 - 2k) $ | Infinite Solutions |
| $ 103 = 7(x + 1) + 5x $ | One Solution |
| $ 9x + 1 = -9(-1 - x) $ | No Solution |
| $ -3n + 3n = 5 $ | No Solution |
| $ -7x + 2 = 2(3 + 2x) - 11x $ | No Solution |
| $ -8(p + 7) = -88 $ | One Solution |
| $ 5n = 5n + 2 $ | No Solution |
| $ 20 + 2x = 2(10 - 3x) + 8x $ | Infinite Solutions |

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