Questions: Determine if the following equation is a conditional equation, an identity, or a contradiction. 2(x-6)+7 x=3 x+6(x-2)

Determine if the following equation is a conditional equation, an identity, or a contradiction. 2(x-6)+7 x=3 x+6(x-2)
Transcript text: Determine if the following equation is a conditional equation, an identity, or a contradiction. \[ 2(x-6)+7 x=3 x+6(x-2) \]
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Solution

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Solution Steps

To determine if the given equation is a conditional equation, an identity, or a contradiction, we need to simplify both sides of the equation and compare them. If the simplified equation is true for all values of \( x \), it is an identity. If it is true for some values of \( x \), it is a conditional equation. If it is never true, it is a contradiction.

Step 1: Simplifying the Left-Hand Side

We start with the left-hand side of the equation: \[ 2(x - 6) + 7x \] Distributing \(2\) gives: \[ 2x - 12 + 7x = 9x - 12 \]

Step 2: Simplifying the Right-Hand Side

Next, we simplify the right-hand side of the equation: \[ 3x + 6(x - 2) \] Distributing \(6\) gives: \[ 3x + 6x - 12 = 9x - 12 \]

Step 3: Comparing Both Sides

Now we compare the simplified left-hand side and right-hand side: \[ 9x - 12 = 9x - 12 \] Since both sides are equal for all values of \(x\), we conclude that the equation is an identity.

Final Answer

The equation is an identity, so the final answer is: \[ \boxed{\text{The equation is an identity.}} \]

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