Problem 47 - Elementary and Intermediate Algebra

Question

Solve each equation. $$ \sqrt{3 x+12}=\sqrt{5 x-12} $$

Step-by-Step Solution

Answer

The solution is \(x = 12\).

Step 1: Eliminate the square roots

To eliminate the square roots, square both sides of the equation. This gives us: \[ ( ext{LHS})^2 = ( ext{RHS})^2 \]. Applying this to the equation, we have:\[ (\sqrt{3x + 12})^2 = (\sqrt{5x - 12})^2 \].This simplifies to:\[ 3x + 12 = 5x - 12 \].

Step 2: Simplify the equation

Now, rearrange the equation to isolate terms involving \(x\) on one side. Subtract \(3x\) from both sides:\[ 12 = 5x - 3x - 12 \],which simplifies to:\[ 12 = 2x - 12 \].

Step 3: Isolate the x variable

To solve for \(x\), first add 12 to both sides of the equation:\[ 12 + 12 = 2x - 12 + 12 \],which simplifies to:\[ 24 = 2x \].Now, divide both sides by 2:\[ x = \frac{24}{2} \],which results in:\[ x = 12 \].

Step 4: Verify the solution

To verify, substitute \(x = 12\) back into the original equation and check:\[ \sqrt{3(12) + 12} = \sqrt{5(12) - 12} \].Simplifying both sides gives:\[ \sqrt{36 + 12} = \sqrt{60 - 12} \],which becomes:\[ \sqrt{48} = \sqrt{48} \].The equation holds true, confirming \(x = 12\) is a valid solution.

Key Concepts

Understanding Square Roots

Before we dive into solving the equation, let's grasp the concept of square roots. A square root of a number is a value that, when multiplied by itself, gives the original number. Think of it as the opposite of squaring a number. For example, the square root of 9 is 3 because 3 multiplied by 3 equals 9.

When you see the symbol \( \sqrt{} \), it denotes the square root. In algebra, the square root function is essential because it allows us to handle equations involving numbers raised to the power of 2. Often, to eliminate the square root in equations like \( \sqrt{3x + 12} = \sqrt{5x - 12} \), we can square both sides, converting the equation into a more straightforward linear form. This step is crucial as it helps in simplifying and solving equations that initially seem complex due to the square roots.

Remember, when you square both sides, ensure all terms are properly simplified to avoid mistakes. This sets the foundation for solving the equation accurately.

Solving for X

Once the square roots in our equation \( \sqrt{3x + 12} = \sqrt{5x - 12} \) are removed by squaring both sides, the equation simplifies to a linear form: \( 3x + 12 = 5x - 12 \). Here is how you can solve for \( x \):

• Start by simplifying, which often involves combining like terms and isolating the variable \( x \).
• Subtract \( 3x \) from both sides to get \( 12 = 2x - 12 \).
• Add 12 to both sides to remove the constant term on the right side, giving you \( 24 = 2x \).
• Finally, divide both sides by 2 to solve for \( x \), resulting in \( x = 12 \).

This step-by-step process, known as isolating the variable, helps you find the unknown value \( x \) in an algebraic equation, effectively transforming complex expressions into something more manageable.

Verification of Solutions

Verification is a critical step in solving algebraic equations. It ensures that the solution you found is indeed correct. For our equation, once we determined \( x = 12 \), it is important to substitute it back into the original equation to verify its accuracy:

• Plug \( x = 12 \) into the original equation: \( \sqrt{3(12) + 12} = \sqrt{5(12) - 12} \).
• Simplify both sides to see if they hold equal values: \( \sqrt{48} = \sqrt{48} \).
• Here, both sides equal, confirming that our solution is correct.

This process, known as verification, not only reaffirms the solution but also boosts your confidence in the methodology used. It is a great step that ensures your calculations and logical steps align with the problem's requirements.