Solve the equation 3x + 8 = 4x - x + 7 and classify it as conditional, contradiction, or identity.

Understand the Problem

The question is asking to solve the equation 3x + 8 = 4x - x + 7 and determine if the equation is conditional, a contradiction, or an identity.

Answer

The equation is a contradiction.

Steps to Solve

Combine Like Terms on the Right Side

Start by simplifying the right side of the equation. Combine $4x$ and $-x$:

$$ 3x + 8 = (4x - x) + 7 $$

This simplifies to:

$$ 3x + 8 = 3x + 7 $$

Subtract $3x$ from Both Sides

Next, move all terms involving $x$ to one side of the equation by subtracting $3x$ from both sides:

$$ 3x + 8 - 3x = 3x + 7 - 3x $$

This simplifies to:

$$ 8 = 7 $$

Analyze the Result

The equation $8 = 7$ is incorrect; it indicates that there is no solution to the original equation. This means the equation is a contradiction.

The type of equation is a contradiction.

More Information

In algebra, equations can be classified as conditional (true for some values of the variable), contradictions (not true for any values), or identities (true for all values of the variable). The given equation simplifies to an impossible statement, confirming it as a contradiction.

Tips

Forgetting to combine like terms on both sides before simplifying can lead to incorrect conclusions.
Misinterpreting the result; if you end up with a false statement like $8=7$, recognize it indicates no solution.

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