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 \(\frac{1}{3}\):

\[
\frac{1}{3} \cdot 3x - \frac{1}{3} \cdot 9 = 18 \Rightarrow x - 3 = 18
\]

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 a false statement.

✘ No solution

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3. \(8(x - 4) = 2(4x - 6)\)

Step 1: Distribute both sides:

Left: \(8x - 32\)
Right: \(8x - 12\)

So:

\[
8x - 32 = 8x - 12
\]

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 the right side:

\[
8 - 15 = -7, \text{ so } x - 7 = x - 7
\]

Step 2: Subtract \(x\) from both sides:

\[
-7 = -7
\]

Always true — this is an identity.

✔ Infinite solutions

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5. \(2x - 4 + 5x = 3x + 7x - 8\)

Step 1: Combine like terms on both sides.

Left: \(2x + 5x - 4 = 7x - 4\)
Right: \(3x + 7x - 8 = 10x - 8\)

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 to eliminate denominator:

\[
2x - 1 = 9
\]

Step 2: Add 1 to both sides:

\[
2x = 10
\]

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 from both sides:

\[
-12 = 4x
\]

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:

\[
6x + 3
\]

Step 2: Distribute right side:

\[
3 \cdot 2x + 3 \cdot 1 = 6x + 3
\]

So:

\[
6x + 3 = 6x + 3
\]

Subtract \(6x\) from both sides:

\[
3 = 3
\]

Always true — identity.

✔ Infinite solutions

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## ✔ Final Answers:

| Problem | Solution Type | Answer (if one solution) |
|---------|---------------------|--------------------------|
| 1 | One solution | \(x = 21\) |
| 2 | No solution | — |
| 3 | No solution | — |
| 4 | Infinite solutions | — |
| 5 | One solution | \(x = \frac{4}{3}\) |
| 6 | One solution | \(x = 5\) |
| 7 | One solution | \(x = -3\) |
| 8 | Infinite solutions | — |

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