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

Let's solve each equation step by step and determine whether it has one solution, no solution, or infinite solutions.

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1.

$$
\frac{1}{3}(3x - 9) = 18
$$

Step 1: Distribute the $\frac{1}{3}$:
$$
\frac{1}{3} \cdot 3x - \frac{1}{3} \cdot 9 = 18 \Rightarrow x - 3 = 18
$$

Step 2: Add 3 to both sides:
$$
x = 21
$$

✔ One solution: $x = 21$

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2.

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

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

✘ This is false — no value of $x$ makes this true.

🚫 No solution

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3.

$$
8(x - 4) = 2(4x - 6)
$$

Step 1: Expand both sides:
$$
8x - 32 = 8x - 12
$$

Step 2: Subtract $8x$ from both sides:
$$
-32 = -12
$$

✘ False statement.

🚫 No solution

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4.

$$
x - 7 = 8 + x - 15
$$

Step 1: Simplify right side:
$$
x - 7 = x - 7
$$

Step 2: Subtract $x$ from both sides:
$$
-7 = -7
$$

✔ True for all values of $x$.

🔁 Infinite solutions

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5.

$$
2x - 4 + 5x = 3x + 7x - 8
$$

Step 1: Combine like terms on both sides:
Left: $7x - 4$
Right: $10x - 8$

So:
$$
7x - 4 = 10x - 8
$$

Step 2: Subtract $7x$ from both sides:
$$
-4 = 3x - 8
$$

Step 3: Add 8 to both sides:
$$
4 = 3x
$$

Step 4: Divide by 3:
$$
x = \frac{4}{3}
$$

✔ One solution: $x = \frac{4}{3}$

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6.

$$
\frac{2x - 1}{3} = 3
$$

Step 1: Multiply both sides by 3:
$$
2x - 1 = 9
$$

Step 2: Add 1:
$$
2x = 10
$$

Step 3: Divide by 2:
$$
x = 5
$$

✔ One solution: $x = 5$

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7.

$$
2x - 1 = 6x + 11
$$

Step 1: Subtract $2x$ from both sides:
$$
-1 = 4x + 11
$$

Step 2: Subtract 11:
$$
-12 = 4x
$$

Step 3: Divide by 4:
$$
x = -3
$$

✔ One solution: $x = -3$

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8.

$$
8x - 2x + 3 = 3(2x + 1)
$$

Step 1: Simplify left side:
$$
6x + 3 = 3(2x + 1)
$$

Step 2: Expand right side:
$$
6x + 3 = 6x + 3
$$

Step 3: Subtract $6x$ and 3 from both sides:
$$
0 = 0
$$

✔ Always true.

🔁 Infinite solutions

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✔ Final Answers:

| Problem | Solution | Type of Solution |
|--------|----------|------------------|
| 1 | $x = 21$ | One solution |
| 2 | No solution | No solution |
| 3 | No solution | No solution |
| 4 | All real numbers | Infinite solutions |
| 5 | $x = \frac{4}{3}$ | One solution |
| 6 | $x = 5$ | One solution |
| 7 | $x = -3$ | One solution |
| 8 | All real numbers | Infinite solutions |

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🔍 Summary:

- One solution: When you get a specific value for $x$.
- No solution: When you get a false statement (e.g., $-1 = 2$).
- Infinite solutions: When you get a true statement that’s always true (e.g., $0 = 0$).

Let me know if you'd like these written out in your worksheet format!