Solve the equation 6x + 5 - 10x = 6 - 4x - 1 and classify the type of solution.

Understand the Problem

The question is asking to solve the equation 6x + 5 - 10x = 6 - 4x - 1 and to determine the type of solution, which can be 'conditional', 'contradiction', or 'identity'.

Answer

The solution is an identity.

Steps to Solve

Combine like terms on the left side

Start with the equation:

$$ 6x + 5 - 10x = 6 - 4x - 1 $$

Combine the terms with $x$ on the left:

$$ (6x - 10x) + 5 = 6 - 4x - 1 $$

This simplifies to:

$$ -4x + 5 = 6 - 4x - 1 $$

Simplify the right side

Now, simplify the right side of the equation by combining like terms:

$$ 6 - 1 = 5 $$

So, the equation now looks like:

$$ -4x + 5 = 5 - 4x $$

Set the two sides equal

The simplified equation is:

$$ -4x + 5 = 5 - 4x $$

Now, notice that both sides are equal.

Identify the type of solution

Since both sides are identical, this implies that the equation holds true for all values of $x$. Thus, the solution is an identity.

The solution to the equation is an identity.

More Information

An identity means that the equation is true for all values of (x). In this case, the equation represents the same line when graphed, indicating an infinite number of solutions.

Tips

Forgetting to combine like terms: Ensure all terms are simplified correctly on both sides before trying to solve.
Not recognizing identities: Some may overlook that if both sides simplify to the same expression, it indicates an identity.

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