Linear Equations: One Solution, No Solution, Infinitely Many ... - Free Printable

Let's solve this Linear Equations Card Sort problem step by step. The goal is to determine how many solutions each equation has: one solution, no solution, or infinitely many solutions.

---

🔍 How to Determine the Number of Solutions:

For linear equations in one variable:

1. One Solution:
After simplifying, you get something like:
$ x = \text{number} $ → unique solution.

2. No Solution:
After simplifying, you get a contradiction like:
$ 0 = 5 $ or $ 3 = -2 $ → no value of $ x $ satisfies it.

3. Infinitely Many Solutions:
After simplifying, you get an identity like:
$ 0 = 0 $ or $ 5 = 5 $ → true for all $ x $.

---

We’ll go through each equation and classify it accordingly.

---

## ✔ ONE SOLUTION (Unique Answer)

These equations simplify to a single value of $ x $.

1. $ 5x + 11 = -4 $

$$
5x = -4 - 11 = -15 \Rightarrow x = -3
$$
✔ One solution

2. $ 6x + 2 = 14 $

$$
6x = 12 \Rightarrow x = 2
$$
✔ One solution

3. $ 18 = -2 + 4x $

$$
4x = 20 \Rightarrow x = 5
$$
✔ One solution

4. $ -9x - 3x = 2(x + 14) $

$$
-12x = 2x + 28 \Rightarrow -14x = 28 \Rightarrow x = -2
$$
✔ One solution

5. $ 3(x + 8) = 15x - 6 + 3x $

$$
3x + 24 = 18x - 6 \Rightarrow 24 + 6 = 15x \Rightarrow 30 = 15x \Rightarrow x = 2
$$
✔ One solution

6. $ 12x - 15 = 8x - 3 $

$$
12x - 8x = -3 + 15 \Rightarrow 4x = 12 \Rightarrow x = 3
$$
✔ One solution

---

## ✘ NO SOLUTION (Contradiction)

These lead to false statements like $ 0 = 5 $

1. $ 7x - 9 = 7x - 3 $

$$
\text{Subtract } 7x \text{ from both sides: } -9 = -3 \quad \text{(False)}
$$
✘ No solution

2. $ 2(x + 8) = 2x + 8 $

$$
2x + 16 = 2x + 8 \Rightarrow 16 = 8 \quad \text{(False)}
$$
✘ No solution

3. $ 4x + x = 8 + 5x $

$$
5x = 8 + 5x \Rightarrow 0 = 8 \quad \text{(False)}
$$
✘ No solution

4. $ x + 5x - 11 = 2(3x + 5) $

$$
6x - 11 = 6x + 10 \Rightarrow -11 = 10 \quad \text{(False)}
$$
✘ No solution

5. $ 5(x + 4) = 5x - 18 $

$$
5x + 20 = 5x - 18 \Rightarrow 20 = -18 \quad \text{(False)}
$$
✘ No solution

6. $ 4x - 16 = 4(x + 4) $

$$
4x - 16 = 4x + 16 \Rightarrow -16 = 16 \quad \text{(False)}
$$
✘ No solution

---

## 🌀 INFINITELY MANY SOLUTIONS (Identity)

These simplify to $ 0 = 0 $ or always true statement.

1. $ -18x = -2(9x) $

$$
-18x = -18x \Rightarrow 0 = 0 \quad \text{(Always true)}
$$
🌀 Infinitely many solutions

2. $ 4x = 5x - x $

$$
4x = 4x \Rightarrow 0 = 0 \quad \text{(Always true)}
$$
🌀 Infinitely many solutions

3. $ -10x + 4x = -6x $

$$
-6x = -6x \Rightarrow 0 = 0 \quad \text{(Always true)}
$$
🌀 Infinitely many solutions

4. $ 6(x - 2) = 3(2x - 4) $

$$
6x - 12 = 6x - 12 \Rightarrow 0 = 0 \quad \text{(Always true)}
$$
🌀 Infinitely many solutions

5. $ 8x + 2x - 6 = 2(5x - 3) $

$$
10x - 6 = 10x - 6 \Rightarrow 0 = 0 \quad \text{(Always true)}
$$
🌀 Infinitely many solutions

6. $ 2(4x + 7) = 13x - 5x + 14 $

$$
8x + 14 = 8x + 14 \Rightarrow 0 = 0 \quad \text{(Always true)}
$$
🌀 Infinitely many solutions

---

✔ Final Classification:

#### One Solution
- $ 5x + 11 = -4 $
- $ 6x + 2 = 14 $
- $ 18 = -2 + 4x $
- $ -9x - 3x = 2(x + 14) $
- $ 3(x + 8) = 15x - 6 + 3x $
- $ 12x - 15 = 8x - 3 $

#### No Solution
- $ 7x - 9 = 7x - 3 $
- $ 2(x + 8) = 2x + 8 $
- $ 4x + x = 8 + 5x $
- $ x + 5x - 11 = 2(3x + 5) $
- $ 5(x + 4) = 5x - 18 $
- $ 4x - 16 = 4(x + 4) $

#### Infinitely Many Solutions
- $ -18x = -2(9x) $
- $ 4x = 5x - x $
- $ -10x + 4x = -6x $
- $ 6(x - 2) = 3(2x - 4) $
- $ 8x + 2x - 6 = 2(5x - 3) $
- $ 2(4x + 7) = 13x - 5x + 14 $

---

✔ Summary:

| Type | Count |
|------|-------|
| One Solution | 6 |
| No Solution | 6 |
| Infinitely Many Solutions | 6 |

Each category has 6 equations, totaling 18 equations.

---

This matches the layout on the worksheet — perfect!

Let me know if you'd like a printable version or explanation for any specific equation!