Questions: Classify the equation as a contradiction, a conditional equation, or an identity. 4 x + 7/3 = (12 x + 7)/3

Classify the equation as a contradiction, a conditional equation, or an identity. 4 x + 7/3 = (12 x + 7)/3
Transcript text: Classify the equation as a contradiction, a conditional equation, or an identity. \[ 4 x+\frac{7}{3}=\frac{12 x+7}{3} \]
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Solution

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Solution Steps

To classify the given equation, we need to simplify and solve it. If the equation has no solution, it is a contradiction. If it has a specific solution, it is a conditional equation. If it is true for all values of \( x \), it is an identity. We will simplify both sides of the equation and solve for \( x \).

Step 1: Simplify the Equation

The given equation is:

\[ 4x + \frac{7}{3} = \frac{12x + 7}{3} \]

To simplify, multiply every term by 3 to eliminate the fraction:

\[ 3(4x) + 3\left(\frac{7}{3}\right) = 12x + 7 \]

This simplifies to:

\[ 12x + 7 = 12x + 7 \]

Step 2: Analyze the Simplified Equation

The simplified equation is:

\[ 12x + 7 = 12x + 7 \]

This equation is true for all values of \( x \), as both sides are identical.

Final Answer

Since the equation is true for all values of \( x \), it is an identity. Therefore, the equation is classified as an identity.

\[ \boxed{\text{Identity}} \]

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