What Equations Are Equivalent To 3(5x+4)=12x+18

What Equations Are Equivalent To 3(5x+4)=12x+18

Understanding equivalent equations is fundamental in algebra, as they allow us to manipulate and solve equations efficiently while maintaining the same solution set. When presented with an equation like 3(5x+4)=12x+18, the question often arises: What other equations are equivalent to it? Equivalence in algebra means that different equations, possibly expressed in different forms, share exactly the same solution(s). This article explores how to identify, create, and verify equations equivalent to 3(5x+4)=12x+18, along with step-by-step methods, common practices, and tips for solving such equations effectively.

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Understanding the Given Equation

Before exploring equivalent equations, it’s essential to understand the structure and solution of the original equation:

Original Equation:

3(5x + 4) = 12x + 18

Step 1: Expand the left side

Using distributive property:

3 5x + 3 4 = 15x + 12

So, the equation becomes:

15x + 12 = 12x + 18

Step 2: Simplify and solve

Subtract 12x from both sides:

15x - 12x + 12 = 18

which simplifies to:

3x + 12 = 18

Subtract 12 from both sides:

3x = 6

Divide both sides by 3:

x = 2

Solution: x = 2

This solution will be the key to verifying whether other equations are equivalent.

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Defining Equivalent Equations

Equivalent equations are different algebraic expressions that share the same solution(s). If two equations are equivalent, any solution that satisfies one will satisfy the other.

Key points about equivalent equations:

• They can be derived from each other through valid algebraic operations such as addition, subtraction, multiplication, or division by non-zero constants.


• They may look different but have the same solution set.


• They maintain the same solutions even after transformations.

• Original: 3(5x + 4) = 12x + 18


• Equivalent: 15x + 12 = 12x + 18 (expanded form)


• Also equivalent: 3(5x + 4) - (12x + 18) = 0 (set equal to zero)

How to Find Equations Equivalent to 3(5x+4)=12x+18

Creating equivalent equations involves applying algebraic operations that do not change the solution set. These include:

1. Addition or Subtraction of the Same Expression

Adding or subtracting the same expression from both sides maintains equivalence.

• Example:

Add 5 to both sides:

\[
3(5x+4) + 5 = 12x + 18 + 5
\]

Simplifies to:

\[
15x + 12 + 5 = 12x + 23
\]

Resulting Equation:

\[
15x + 17 = 12x + 23
\]

Since adding the same number to both sides preserves the solution set, this new equation is equivalent.

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2. Multiplication or Division by Non-zero Constants

Multiplying or dividing both sides of the equation by a non-zero number also preserves equivalence.

• Example:

\[
15x + 12 = 12x + 18
\]

Divide both sides by 3:

\[
\frac{15x}{3} + \frac{12}{3} = \frac{12x}{3} + \frac{18}{3}
\]

Simplifies to:

\[
5x + 4 = 4x + 6
\]

This new equation is equivalent to the original because we divided both sides by 3, a non-zero constant.

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3. Re-arranging Terms (Adding or Subtracting Terms)

You can also rearrange the equation to different forms by moving terms around, as long as you perform the same operation on both sides.

• Example:

\[
15x + 12 = 12x + 18
\]

Subtract 12x from both sides:

\[
3x + 12 = 18
\]

Subtract 12 from both sides:

\[
3x = 6
\]

This is an equivalent simplified form.

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Step-by-Step Approach to Generate Equivalent Equations

To systematically find equations equivalent to the original, follow these steps:

• Start from the original equation: 3(5x + 4) = 12x + 18
• Expand or simplify as needed: 15x + 12 = 12x + 18
1. Apply algebraic operations:
   • Add or subtract constants to both sides
   • Multiply or divide both sides by a non-zero constant
   • Rearrange terms to isolate variables
• Verify the solution set: Solve the transformed equation to confirm x=2 is still valid
• Repeat with different operations: Generate multiple equivalent forms for practice and understanding

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Examples of Equations Equivalent To 3(5x+4)=12x+18

Here are several equations that are equivalent to the original:

Example 1: Simplified form

\[
3x = 6
\]

Derived by subtracting 12 from both sides and dividing by 3.

Example 2: Expanded form

\[
15x + 12 = 12x + 18
\]

Starting from the original and expanding.

Example 3: Re-arranged form

\[
15x + 12 - 12x - 18 = 0
\]

which simplifies to:

\[
3x - 6 = 0
\]

Adding all terms to one side.

Example 4: Equation in terms of zero

\[
3(5x + 4) - (12x + 18) = 0
\]

Subtracting the right side from the left.

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Verifying Equivalence of Equations

It is crucial to verify that the equations are truly equivalent, particularly after transformations.

Methods for verification include:

• Solving both equations: If both yield the same solution(s), they are equivalent.


• Substituting the solution(s): Plug solutions into each equation to check if they satisfy both.


• Using algebraic equivalence rules: Confirm operations performed are valid and preserve solutions.

• For the equations:

and

\[
3x = 6
\]

test x=2:

\[
3(52 + 4) = 122 + 18 \rightarrow 3(10 + 4) = 24 + 18 \rightarrow 314=42 \rightarrow 42=42
\]

and

\[
32=6 \rightarrow 6=6
\]

Both true, confirming equivalence.

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Common Mistakes To Avoid When Working With Equivalent Equations

While manipulating equations to find equivalents, avoid these frequent errors:

• Dividing by zero: Never divide both sides by a variable expression that could be zero; always check for extraneous solutions.


• Changing the solution set unintentionally: Operations like multiplying both sides by a negative number require attention to inequality signs (not relevant here but important in inequalities).


• Incorrectly adding or subtracting terms: Make sure to perform the same operation on both sides.


• Misapplying the distributive property: Always expand expressions properly before transforming.

Practical Tips and Best Practices

• Always verify the solutions after transforming equations.
• Keep track of operations performed to ensure they are valid and reversible.
• Practice transforming equations in multiple ways to build flexibility.
• Use substitution to verify that different forms are equivalent.
• Understand the properties of equality: Additive, multiplicative, and distributive properties are key to safe transformations.

Conclusion

Equations equivalent to 3(5x+4)=12x+18 can be generated through various algebraic manipulations such as expansion, factoring, adding or subtracting constants, and dividing or multiplying by non-zero constants.