Problem 64 - Algebra for College Students

Question

For the indicated variable. Assume that all letters represent positive numbers. \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1\) for \(y\)

Step-by-Step Solution

Answer

\( y = \pm \sqrt{\frac{b^{2}x^{2}}{a^{2}} - b^{2}} \)

Step 1: Rearrange the Equation

Start with the original equation: \( \frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1 \). To solve for \( y \), first move the term with \( y^{2} \) to the other side by adding \( \frac{y^{2}}{b^{2}} \) to both sides: \( \frac{x^{2}}{a^{2}} = 1 + \frac{y^{2}}{b^{2}} \).

Step 2: Isolate the Term with y

To get the term with \( y^{2} \) alone, subtract \( 1 \) from both sides: \( \frac{x^{2}}{a^{2}} - 1 = \frac{y^{2}}{b^{2}} \).

Step 3: Clear the Fraction

Multiply both sides by \( b^{2} \) to eliminate the fraction on the right: \( b^{2} \left( \frac{x^{2}}{a^{2}} - 1 \right) = y^{2} \).

Step 4: Simplify

Distribute \( b^{2} \) to both terms inside the parentheses: \( y^{2} = \frac{b^{2}x^{2}}{a^{2}} - b^{2} \).

Step 5: Solve for y

Take the square root of both sides to solve for \( y \). Remember to include the positive and negative roots: \( y = \pm \sqrt{\frac{b^{2}x^{2}}{a^{2}} - b^{2}} \).

Key Concepts

Equation Solving

Equation solving involves finding the unknown value that makes the equation true. In this exercise, we're solving for the variable \( y \) in the given equation. The goal is to isolate \( y \) on one side of the equation, which involves a few steps:

• First, rearrange the terms to move all instances of \( y \) to one side.
• Next, manipulate the equation to separate the term containing \( y \).
• Then, simplify the equation further to express \( y \) in terms of other known quantities.

These steps require knowledge of algebraic operations such as adding, subtracting, multiplying, and dividing terms, as well as the ability to understand and apply the concept of the square root when solving for \( y \). Remember, taking the square root introduces both positive and negative solutions, since both can satisfy the equation when squared.

Variables in Algebra

In algebra, variables are symbols that represent unknown values. They are often denoted by letters like \( x \), \( y \), and \( a \). Variables allow us to write general expressions and equations that can describe a wide range of possible scenarios. Variables play a crucial role in forming equations, enabling us to solve for the unknowns. Here is how variables work:

• They can be used to represent numbers that change or are not yet specified.
• Algebraic manipulation of variables involves performing operations like addition, subtraction, and factoring to simplify expressions.
• Solving for a variable means finding the value(s) that make an equation true.

By using variables, complex real-world problems can often be transformed into mathematical equations that are more manageable to solve.

Mathematical Expressions

Mathematical expressions are combinations of numbers, variables, and operation signs that represent a value. They are the building blocks of equations. Understanding how to work with mathematical expressions is key to solving algebraic equations. In the exercise, expressions are used to represent relationships between different quantities. Let's break down what you need to know:

• Expressions can include constants (fixed values), coefficients (numbers multiplying variables), and exponents (indicating powers).
• To manipulate expressions, you often need to use distributive properties and combine like terms.
• Simplification is an important step, often achieved by performing operations like addition or subtraction within the expression.

When working with expressions, the ability to apply these principles can greatly simplify your work in algebra, making it easier to solve problems and understand the relationships between variables.