Equivalent Equations

Article Contents

1. What are Equivalent Equations?
2. Rules for Creating Equivalent Equations
3. Visual Examples
4. Practice Problems
5. Interactive Exercises

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1. What are Equivalent Equations? {#what-are-equivalent-equations}

Equivalent equations are equations that have the same solution(s).

If $x=3$ makes equation A true, and $x=3$ also makes equation B true, then A and B are equivalent.

Example

These three equations are all equivalent (they all have $x=5$ as their solution):

Equation	Verification
$x+2=7$	$5+2=7✓$
$x=7-2$	$x=5✓$
$2x=10$	$2(5)=10✓$

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2. Rules for Creating Equivalent Equations {#rules-for-creating-equivalent-equations}

The balance principle gives us these rules:

Rule 1: Add the Same Number to Both Sides

$$x-3=7\iff x-3+3=7+3\iff x=10$$

Rule 2: Subtract the Same Number from Both Sides

$$x+5=12\iff x+5-5=12-5\iff x=7$$

Rule 3: Multiply Both Sides by the Same Number

$$x/2=6\iff x/2\times 2=6\times 2\iff x=12$$

Rule 4: Divide Both Sides by the Same Number (non-zero)

$$3x=15\iff 3x/3=15/3\iff x=5$$

Rule 5: Swap Sides (Reflexive Property)

$$x+2=7\iff 7=x+2$$

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3. Visual Examples {#visual-examples}

Transformation Chain

Starting with: $x+4=12$

```

Step 1: x + 4 = 12

Step 2: x = 12 - 4 (subtract 4 from both sides)

Step 3: x = 8 (calculate)

```

Each step produces an equivalent equation.

Multiple Transformations

Starting with: $2x+3=11$

```

Step 1: 2x + 3 = 11

Step 2: 2x = 11 - 3 (subtract 3 from both sides)

Step 3: 2x = 8 (calculate)

Step 4: x = 8/2 (divide both sides by 2)

Step 5: x = 4 (calculate)

```

All equations in this chain are equivalent!

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4. Practice Problems {#practice-problems}

Problem 1

Which equation is equivalent to $x+6=14$?

• A) $x=14+6$
• B) $x=14-6$
• C) $x=14\times 6$
• D) $x=14÷6$

Answer: B ($x=14-6=8$)

Problem 2

Which transformation of $3x=21$ gives an equivalent equation?

• A) Add 3 to both sides
• B) Subtract 3 from both sides
• C) Divide both sides by 3
• D) Multiply both sides by 3

Answer: C ($x=7$)

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5. Interactive Exercises {#interactive-exercises}

Test your understanding of equivalent equations:

• Equivalent Equations Quiz - Multiple choice identification

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