Kaylee writes the equation 6x + 12 = 2(3x + 6). Can you find the number of solutions this equation has without solving for x? Explain.

Understand the Problem

The question asks whether one can determine the number of solutions for the equation 6x + 12 = 2(3x + 6) without actually solving for x. To address this, we might analyze the equation's structure and simplify both sides to understand if they represent the same linear relationship or if there's a point of intersection.

Answer

The equation has infinitely many solutions.

Steps to Solve

Simplify the equation

First, simplify the right side of the equation:

$$ 2(3x + 6) = 6x + 12 $$

This shows that both sides of the equation are identical.

Compare both sides of the equation

Now the equation reads:

$$ 6x + 12 = 6x + 12 $$

This demonstrates that both sides represent the same linear expression.

Determine the number of solutions

Since both sides of the equation are the same, this means that any value for $x$ will satisfy the equation. Thus, the equation has infinitely many solutions.

The equation has infinitely many solutions.

More Information

When two expressions are identical, any value substituted for the variable will create a true statement. For example, substituting $x = 0$, $x = 1$, or any real number will satisfy the equation.

Tips

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