This algebra worksheet breaks down the three types of equation solutions with clear definitions, worked examples, and practice problems to help students master linear equations.

You’ve uploaded an image showing a chart that defines three types of solutions for linear equations:

- One Solution: The equation has one unique solution.
*Example:* `3x - 5 = x + 10`

- No Solution:
- There is no real number that can make the equation true.
- The variables cancel out.
- The final equation is not true.
*Example:* `2x - 4 = 2x + 3`

- Infinite Solutions:
- Any real number can make the equation true.
- The variables cancel out.
- The final equation is true.
*Example:* `5(x - 1) = 5x - 5`

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Below the chart, there are 4 equations to solve and classify:

> Examples: Determine if each equation has one, no, or infinite solutions.

Let’s solve each one step by step.

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1. 3x + 5 = 3x - 2

Step 1: Subtract `3x` from both sides:

```
3x + 5 - 3x = 3x - 2 - 3x
→ 5 = -2
```

Result: `5 = -2` → This is false.

✔ Conclusion: NO SOLUTION

*(Variables canceled out, final equation is false — matches “No Solution” definition.)*

---

2. 2x - 12 = 8x

Step 1: Subtract `2x` from both sides:

```
2x - 12 - 2x = 8x - 2x
→ -12 = 6x
```

Step 2: Divide both sides by 6:

```
x = -12 / 6 = -2
```

✔ Conclusion: ONE SOLUTION (`x = -2`)

*(We found a unique value for x — matches “One Solution” definition.)*

---

3. 2(5x - 4) = 10x - 8

Step 1: Distribute the 2 on the left:

```
10x - 8 = 10x - 8
```

Step 2: Subtract `10x` from both sides:

```
10x - 8 - 10x = 10x - 8 - 10x
→ -8 = -8
```

Result: `-8 = -8` → This is true.

✔ Conclusion: INFINITE SOLUTIONS

*(Variables canceled out, final equation is true — matches “Infinite Solutions” definition.)*

---

4. 8x - 4x = 18

Step 1: Combine like terms on the left:

```
4x = 18
```

Step 2: Divide both sides by 4:

```
x = 18/4 = 9/2 = 4.5
```

✔ Conclusion: ONE SOLUTION (`x = 4.5`)

*(Unique solution found — matches “One Solution” definition.)*

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## ✔ Final Answers:

1. No Solution
2. One Solution
3. Infinite Solutions
4. One Solution

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## 💡 Summary Tip:

When solving linear equations:

- If you end up with a false statement (like `5 = -2`) → No Solution
- If you end up with a true statement (like `0 = 0` or `-8 = -8`) → Infinite Solutions
- If you get a specific value for x → One Solution

This method works every time!