Questions: Ex 3: Identify each equation as a conditional equation, a contradiction, or an identity. a) 6x-3=2(3x-2) b) 6x-3=2(3x-2)+1

Ex 3: Identify each equation as a conditional equation, a contradiction, or an identity. a) 6x-3=2(3x-2) b) 6x-3=2(3x-2)+1
Transcript text: Ex 3: Identify each equation as a conditional equation, a contradiction, or an identity. a) $6 x-3=2(3 x-2)$ b) $6 x-3=2(3 x-2)+1$
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Solution

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Solution Steps

To determine whether each equation is a conditional equation, a contradiction, or an identity, we need to simplify both sides of the equation and compare the results.

  1. Simplify both sides of the equation.
  2. Compare the simplified forms:
    • If the simplified forms are identical, the equation is an identity.
    • If the simplified forms result in a false statement (e.g., $0 = 1$), the equation is a contradiction.
    • Otherwise, the equation is conditional.
Step 1: Simplify the Equation (a)

Given equation: \[ 6x - 3 = 2(3x - 2) \]

First, distribute the 2 on the right-hand side: \[ 6x - 3 = 6x - 4 \]

Step 2: Compare Both Sides of the Equation (a)

Subtract \(6x\) from both sides: \[ 6x - 3 - 6x = 6x - 4 - 6x \] \[ -3 = -4 \]

This is a contradiction because \(-3\) does not equal \(-4\).

Final Answer for (a)

\(\boxed{\text{Contradiction}}\)

Step 3: Simplify the Equation (b)

Given equation: \[ 6x - 3 = 2(3x - 2) + 1 \]

First, distribute the 2 on the right-hand side: \[ 6x - 3 = 6x - 4 + 1 \]

Combine like terms on the right-hand side: \[ 6x - 3 = 6x - 3 \]

Step 4: Compare Both Sides of the Equation (b)

Subtract \(6x\) from both sides: \[ 6x - 3 - 6x = 6x - 3 - 6x \] \[ -3 = -3 \]

This is an identity because \(-3\) equals \(-3\) for all values of \(x\).

Final Answer for (b)

\(\boxed{\text{Identity}}\)

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