Algebra 1 Worksheets | Equations Worksheets - Free Printable

Let's solve each equation step by step and determine the type of solution (one, infinite, or no solutions). We'll go through all 12 problems.

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Problem 1:

Equation:
$$
-68 + 30k = 10(3k - 7) + 2
$$

Step 1: Expand the right-hand side:
$$
10(3k - 7) + 2 = 30k - 70 + 2 = 30k - 68
$$

Now the equation is:
$$
-68 + 30k = 30k - 68
$$

Step 2: Subtract $30k$ from both sides:
$$
-68 = -68
$$

This is always true.
✔ Answer: Infinite solutions

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Problem 2:

Equation:
$$
48(m + 3) - 102 = 6(8m + 7)
$$

Step 1: Expand both sides:
Left: $48m + 144 - 102 = 48m + 42$
Right: $48m + 42$

So:
$$
48m + 42 = 48m + 42
$$

Subtract $48m + 42$ from both sides:
$$
0 = 0
$$

✔ Answer: Infinite solutions

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Problem 3:

Equation:
$$
21 + 3h = 3(h + 7)
$$

Step 1: Expand right side:
$$
3h + 21
$$

So:
$$
21 + 3h = 3h + 21
$$

Subtract $3h + 21$:
$$
0 = 0
$$

✔ Answer: Infinite solutions

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Problem 4:

Equation:
$$
4 = -8z - 7 + 8z
$$

Step 1: Combine like terms on the right:
$$
-8z + 8z = 0 \Rightarrow 4 = -7
$$

✘ This is false.
✔ Answer: No solution

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Problem 5:

Equation:
$$
-8n + 7 + 8n = 7
$$

Simplify left:
$$
(-8n + 8n) + 7 = 0 + 7 = 7
$$

So:
$$
7 = 7
$$

✔ Answer: Infinite solutions

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Problem 6:

Equation:
$$
3 + 4v = 6(5 - 7v)
$$

Step 1: Expand right:
$$
30 - 42v
$$

Now:
$$
3 + 4v = 30 - 42v
$$

Step 2: Add $42v$ to both sides:
$$
3 + 46v = 30
$$

Subtract 3:
$$
46v = 27
$$

Divide:
$$
v = \frac{27}{46}
$$

✔ Answer: One solution → $ v = \frac{27}{46} $

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Problem 7:

Equation:
$$
2(-5f - 9) = -22 - 10f
$$

Step 1: Expand left:
$$
-10f - 18
$$

So:
$$
-10f - 18 = -22 - 10f
$$

Add $10f$ to both sides:
$$
-18 = -22
$$

✘ False
✔ Answer: No solution

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Problem 8:

Equation:
$$
-66 = 9(-7 - 8r) + 72r
$$

Step 1: Expand right:
$$
9(-7) + 9(-8r) = -63 - 72r
$$

Then add $72r$:
$$
-63 - 72r + 72r = -63
$$

So:
$$
-66 = -63
$$

✘ False
✔ Answer: No solution

---

Problem 9:

Equation:
$$
6(t - 10) = -60 + 6t
$$

Step 1: Expand left:
$$
6t - 60
$$

So:
$$
6t - 60 = -60 + 6t
$$

Subtract $6t - 60$:
$$
0 = 0
$$

✔ Answer: Infinite solutions

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Problem 10:

Equation:
$$
12 = -6(9t - 2) + 54t
$$

Step 1: Expand right:
$$
-54t + 12 + 54t = 12
$$

So:
$$
12 = 12
$$

✔ Answer: Infinite solutions

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Problem 11:

Equation:
$$
8(5 - 4c) + 25c + 7c = 10
$$

Step 1: Expand:
$$
40 - 32c + 25c + 7c = 10
$$

Combine like terms:
$$
40 + (-32c + 25c + 7c) = 40 + 0c = 40
$$

So:
$$
40 = 10
$$

✘ False
✔ Answer: No solution

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Problem 12:

Equation:
$$
8 + u = 5u - 4(u + 9)
$$

Step 1: Expand right:
$$
5u - 4u - 36 = u - 36
$$

So:
$$
8 + u = u - 36
$$

Subtract $u$ from both sides:
$$
8 = -36
$$

✘ False
✔ Answer: No solution

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✔ Final Answers Summary:

| Problem | Solution Type | Answer |
|--------|---------------------|--------|
| 1 | Infinite solutions | All real numbers |
| 2 | Infinite solutions | All real numbers |
| 3 | Infinite solutions | All real numbers |
| 4 | No solution | No solution |
| 5 | Infinite solutions | All real numbers |
| 6 | One solution | $ v = \frac{27}{46} $ |
| 7 | No solution | No solution |
| 8 | No solution | No solution |
| 9 | Infinite solutions | All real numbers |
| 10 | Infinite solutions | All real numbers |
| 11 | No solution | No solution |
| 12 | No solution | No solution |

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🔍 Key Observations:

- Infinite solutions: When both sides simplify to the same expression (e.g., $5x + 2 = 5x + 2$).
- No solution: When you get a contradiction (e.g., $5 = 0$).
- One solution: When variables cancel out and you get a specific value.

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