Special Cases: Number of Solutions

Article Contents

1. Three Possibilities
2. One Solution
3. No Solution
4. Infinite Solutions
5. How to Identify Each Case
6. Interactive Exercises

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1. Three Possibilities {#three-possibilities}

When solving a linear equation, exactly one of these happens:

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2. One Solution {#one-solution}

This is the "normal" case.

General Form: $ax=b$ where $a\ne 0$

Example

Solve: $2x+3=11$

Step 1: 2x = 11 - 3 = 8
Step 2: x = 8/2 = 4

The equation has exactly one solution: $x=4$.

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3. No Solution {#no-solution}

This happens when solving leads to a false statement.

General Form: $0x=b$ where $b\ne 0$

Example

Solve: $x+2=x+5$

Step 1: Subtract x from both sides
        x - x + 2 = x - x + 5
Step 2: Simplify
        2 = 5  ❌ FALSE!

This equation has no solution.

Why?

The left side will always be 2 more than the right side, no matter what $x$ is.

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4. Infinite Solutions {#infinite-solutions}

This happens when solving leads to a true statement.

General Form: $0x=0$

Example

Solve: $2x+4=2(x+2)$

Step 1: Expand right side
        2x + 4 = 2x + 4
Step 2: Subtract 2x from both sides
        4 = 4  ✓ TRUE!

This equation has infinitely many solutions (any $x$ works).

Why?

$2(x+2)$ always equals $2x+4$, no matter what $x$ is.

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5. How to Identify Each Case {#how-to-identify-each-case}

After simplifying, look at what's left:

Quick Reference

Equation looks like...     Result:
x = 5                       ONE SOLUTION
3 = 7                       NO SOLUTION  
8 = 8                       INFINITE SOLUTIONS

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Interactive Exercises

• Counting Solutions - Practice identifying the number of solutions

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