Problem 44 - Elementary Algebra

Question

For the following problems, solve for the indicated variable. $$ 9 y^{2}=27 x^{2} z^{4}, \text { for } y $$

Step-by-Step Solution

Answer

Answer: \(y = \sqrt{3x^2z^4}\)

Step 1: Write the given equation

Write down the given equation: $$ 9y^2 = 27x^2 z^4 $$

Step 2: Divide both sides by 9

In order to isolate the term \(y^2\), we should divide both sides of the equation by 9: $$ \frac{9y^2}{9} = \frac{27x^2 z^4}{9} $$ Simplify the equation: $$ y^2 = 3x^2z^4 $$

Step 3: Take the square root of both sides

To get \(y\) alone, take the square root of both sides of the equation: $$ \sqrt{y^2} = \sqrt{3x^2z^4} $$ Simplify the equation further: $$ y = \sqrt{3x^2z^4} $$ Now, we've successfully solved for the variable \(y\) in terms of \(x\) and \(z\): $$ y = \sqrt{3x^2z^4} $$

Key Concepts

Algebraic Manipulation

Algebraic manipulation is a fundamental skill in solving equations. It involves a series of strategic steps to transform equations in order to isolate and solve for a given variable. By rearranging terms and performing operations, you can simplify equations and make them more manageable.In our provided exercise, we start with the equation:

• \(9y^2 = 27x^2 z^4\)

Here, our goal is to solve for the variable \(y\). To achieve this, algebraic manipulation is employed to isolate \(y^2\). This involves dividing both sides of the equation by 9, which simplifies the process. The division allows us to work with more manageable numbers and clearly track our progress toward the solution. Key points include:

• Identifying terms you need to move or simplify.
• Performing operations on both sides to keep the equation balanced.
• Simplifying fractions and terms when possible.

By diligently applying these steps, algebraic manipulation can turn a complex equation into a simpler one, which is essential for finding solutions.

Square Roots

Taking the square root is an essential part of solving equations involving squared terms. When we see equations like \( y^2 = a \), one must use the concept of square roots to solve for \(y\).In the example problem, after dividing by 9, we have:

• \(y^2 = 3x^2z^4\)

To solve for \(y\), we take the square root of both sides. This is a crucial step because it helps us remove the exponent on \(y^2\). Mathematically, it can be expressed as:

• \(y = \sqrt{3x^2z^4}\)

Taking a square root essentially reverses the squaring operation, allowing us to "simplify" the variable under the radical. It's important to recognize when the square root is applicable and to ensure the equation is set up for this operation.Some important reminders:

• Square roots involve ± (positive and negative) results, but context or further constraints should guide which to choose.
• Be aware of when the square root resolves to cleaner or simpler forms, especially with perfect squares.

Understanding the square root operation lets you confidently dismantle squared terms and push towards solving for individual variables.

Variables

Variables are symbols that represent unknown values in algebraic equations. They are placeholders that allow us to set up and solve mathematical problems without knowing all the quantities involved upfront. Commonly used variables include letters like \(x\), \(y\), and \(z\).In our equation:

• \(9y^2 = 27x^2z^4\)

\(y\), \(x\), and \(z\) are all variables, and the task is to solve for one variable in terms of the others. Here, we focus on isolating \(y\). This showcases a fundamental aspect of working with variables: clarity and organization. Keeping track of each variable and its role in the equation is essential.Working with variables often requires:

• Understanding their interaction within the equation.
• Recognizing constants and coefficients attached to variables.
• Employing operations to neatly isolate or simplify variables.

Mastering the use of variables is crucial in algebra, as it prepares you to handle larger, more complex systems of equations that may involve multiple unknowns.