Problem 40 - Algebra 2 and Trigonometry

Question

In \(39-42,\) solve and check each equation. $$ 12 y+\sqrt{32}=\sqrt{200} $$

Step-by-Step Solution

Answer

The solution is \( y = \frac{\sqrt{2}}{2} \).

Step 1: Simplify the Square Roots

Start by simplifying the square roots in the equation. We can express each square root radical as a simpler form using perfect squares.- For \( \sqrt{32} \), note that it is equivalent to \( \sqrt{16 \times 2} = 4\sqrt{2} \).- For \( \sqrt{200} \), it is equivalent to \( \sqrt{100 \times 2} = 10\sqrt{2} \). Thus, the equation becomes: \[ 12y + 4\sqrt{2} = 10\sqrt{2} \]

Step 2: Isolate the Term with Variable

Rearrange the equation to isolate the term that contains the variable \( y \). To do this, subtract \( 4\sqrt{2} \) from both sides.\[ 12y + 4\sqrt{2} - 4\sqrt{2} = 10\sqrt{2} - 4\sqrt{2} \] This simplifies to: \[ 12y = 6\sqrt{2} \]

Step 3: Solve for the Variable

Solve for \( y \) by dividing both sides of the equation by 12. \[ y = \frac{6\sqrt{2}}{12} = \frac{\sqrt{2}}{2} \]

Step 4: Check the Solution

To verify our solution, substitute \( y = \frac{\sqrt{2}}{2} \) back into the original equation:\[ 12\left(\frac{\sqrt{2}}{2}\right) + \sqrt{32} = \sqrt{200} \]Calculate the left-hand side:\[ 12 \times \frac{\sqrt{2}}{2} = 6\sqrt{2} \]Thus, the left-hand side becomes:\[ 6\sqrt{2} + 4\sqrt{2} = 10\sqrt{2} \]Since both sides equal \( 10\sqrt{2} \), the solution \( y = \frac{\sqrt{2}}{2} \) is correct.

Key Concepts

Solving Equations

Solving equations is a key skill in Algebra 2. The goal is to find the value of the variable that makes the equation true. An equation represents a balance, like a seesaw. What you do to one side of the equation, you must do to the other to maintain this balance. In the given problem, the equation is \( 12y + \sqrt{32} = \sqrt{200} \). The journey to solving it involves simplifying and manipulating the equation step by step until you isolate the variable on one side. Consistently applying algebraic rules and operations, like addition, subtraction, and division, helps in maintaining balance and arriving at the correct solution. Remember that each step in manipulating the equation brings you closer to finding the variable's value.

Simplifying Square Roots

Simplifying square roots can seem challenging, but it's just about breaking down numbers into their prime factors. When you simplify a square root, look for perfect square factors. For instance, in our equation, \( \sqrt{32} \) and \( \sqrt{200} \) need simplification.

• \( \sqrt{32} = \sqrt{16 \times 2} = 4\sqrt{2} \), because 16 is a perfect square.
• \( \sqrt{200} = \sqrt{100 \times 2} = 10\sqrt{2} \).

Breaking down square roots into simpler forms makes the equations easier to manage. By simplifying square roots into terms involving \( \sqrt{2} \), we transform the original equation into something more straightforward: \( 12y + 4\sqrt{2} = 10\sqrt{2} \). This lays the groundwork for solving the problem by comparing coefficients of like terms.

Variable Isolation

Variable isolation involves maneuvering an equation until the variable stands alone on one side. This step is critical for pinpointing your final answer. Once the equation \( 12y + 4\sqrt{2} = 10\sqrt{2} \) is simplified, your task is to shift every number that doesn't contain the variable to the opposite side.
To do this, subtract \( 4\sqrt{2} \) from both sides:

• This gives \( 12y = 6\sqrt{2} \), making it clear you have isolated \( y \).

Now, to completely isolate \( y \), divide both sides by 12, resulting in \( y = \frac{6\sqrt{2}}{12} = \frac{\sqrt{2}}{2} \). This step shows the solution for \( y \) and demonstrates how each action simplifies the equation further, ensuring that the variable's value is expressed in the simplest terms possible.

Checking Solutions

Checking your solution is a vital part of solving algebraic equations to ensure correctness. After finding that \( y = \frac{\sqrt{2}}{2} \), substitute this back into the original equation to verify it holds:

• Left side: \( 12\left(\frac{\sqrt{2}}{2}\right) + \sqrt{32} = 6\sqrt{2} + 4\sqrt{2} = 10\sqrt{2} \).
• Right side: \( \sqrt{200} = 10\sqrt{2} \).

Both sides of the equation should match if the solution is correct, which they do.
Checking solutions helps catch errors and builds confidence. It also reinforces understanding by retracing the steps in reverse, proving that your variable manipulation was consistent and correct. If they don't match, you might need to revisit the steps to discover where an error might have occurred.