Types of Solutions Notes and Worksheets - Lindsay Bowden - Free Printable

It looks like you've uploaded an image containing a math worksheet about solving equations and determining whether they have one solution, no solution, or infinite solutions. Since I can't view images directly, I’ll help you solve the problems based on the text you provided.

Let’s go through each equation step by step and determine the number of solutions.

---

Guidelines from the Chart:

| Solution Type | Definition | Example |
|---------------|-----------|--------|
| One Solution | The equation has one unique solution. | $ 3x - 5 = x + 10 $ |
| No Solution | There is no real number that can make the equation true. The variables cancel out, and the final equation is not true. | $ 2x - 4 = 2x + 3 $ |
| Infinite Solutions | Any real number can make the equation true. The variables cancel out, and the final equation is true. | $ 5(x - 1) = 5x - 5 $ |

---

Now let's solve each of the given equations:

---

1. $ 3x + 5 = 3x - 2 $

Step 1: Subtract $ 3x $ from both sides:
$$
3x + 5 - 3x = 3x - 2 - 3x
$$
$$
5 = -2
$$

This is false. No value of $ x $ can make this true.

✔ Answer: No Solution

> Because the variables canceled out and we got a false statement.

---

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:
$$
x = -2
$$

✔ Answer: One Solution (specifically, $ x = -2 $)

> Only one value satisfies the equation.

---

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

Step 1: Expand the left side:
$$
2 \cdot 5x - 2 \cdot 4 = 10x - 8
$$
$$
10x - 8 = 10x - 8
$$

Step 2: Subtract $ 10x $ from both sides:
$$
-8 = -8
$$

This is true for all values of $ x $.

✔ Answer: Infinite Solutions

> The variables canceled out and we got a true statement.

---

4. $ 8x - 4x = 18 $

Step 1: Combine like terms:
$$
4x = 18
$$

Step 2: Divide both sides by 4:
$$
x = \frac{18}{4} = \frac{9}{2} = 4.5
$$

✔ Answer: One Solution ($ x = 4.5 $)

---

✔ Final Answers:

| Equation | Number of Solutions |
|--------|---------------------|
| 1. $ 3x + 5 = 3x - 2 $ | No Solution |
| 2. $ 2x - 12 = 8x $ | One Solution |
| 3. $ 2(5x - 4) = 10x - 8 $ | Infinite Solutions |
| 4. $ 8x - 4x = 18 $ | One Solution |

---

🔍 Summary Tips:

- If variables cancel and you get a false statement (like $ 5 = -2 $), → No Solution
- If variables cancel and you get a true statement (like $ -8 = -8 $), → Infinite Solutions
- If you can solve for a specific value of $ x $, → One Solution

Let me know if you'd like these explained with diagrams or visual steps!