Problem 84 - Algebra for College Students

Question

Solve each equation for the indicated variable. \(3 a y^{2}=b y\) for \(y\)

Step-by-Step Solution

Answer

\(y = \frac{b}{3a}\) assuming \(y \neq 0\) and \(a \neq 0\).

Step 1: Simplify the Equation

To simplify the given equation \(3ay^2 = by\), we need to isolate the common factor on both sides. Start by dividing both sides by \(y\), assuming \(y eq 0\) in order to avoid division by zero. This gives us \(3ay = b\).

Step 2: Solve for y

Now that we have the simplified equation \(3ay = b\), solve for \(y\) by dividing both sides by \(3a\). This results in \(y = \frac{b}{3a}\), and assuming \(a eq 0\) to avoid division by zero.

Key Concepts

Understanding Variables in Equations

In algebra, variables are symbols that represent unknown or changeable values. They play a crucial role in forming equations, where different mathematical operations are applied to these variables. In the exercise, we have two variables, \(a\) and \(y\), within the equation \(3ay^2 = by\).

• \(a\) is a variable that acts as a constant in this context, which means its value does not change throughout the equation's solution.
• \(y\) is the variable we need to solve for, making it the main focus.

Variables allow us to generalize mathematical relationships and solve complex problems by substituting different numbers to find which values satisfy the equation. In this case, our task is to find the specific value of \(y\) that makes the equation true.

Simplification of Equations

Simplification is an essential process in algebra, aimed at reducing complexities and making problems easier to solve. In our exercise, the equation \(3ay^2 = by\) can initially seem complicated because of the squared term and multiple variables.

• We start simplification by dividing both sides of the equation by \(y\), reducing the problem to \(3ay = b\).

This process helps in removing unnecessary parts from the equation without altering its meaning or solution. Simplification often involves factoring or cancelling common terms, making the equation more straightforward and focused on the relatable variables.

Avoiding Division by Zero

Division by zero is a concept that must be carefully avoided in mathematics because it is undefined. Trying to divide by zero does not produce a meaningful number, and in algebra, this can lead to false or misleading solutions. In our exercise:

• When dividing both sides of the equation \(3ay^2 = by\) by \(y\), we ensure that \(y eq 0\). This is critical because division by zero is impossible and would disrupt our calculations.
• Similarly, in the step where we solve for \(y\) by dividing by \(3a\), we assume \(a eq 0\) to keep the operation valid.

Being mindful of situations where division by zero might occur helps in developing correct solutions and maintaining the integrity of mathematical operations.

Algebraic Manipulation Techniques

Algebraic manipulation involves rearranging and adjusting equations to isolate desired variables or simplify expressions. This technique is crucial in solving equations effectively. In the exercise given, algebraic manipulation is employed as follows:

• Initially, dividing the original equation \(3ay^2 = by\) by \(y\) simplifies it to \(3ay = b\).
• Next, we perform another division by \(3a\) to solve for \(y\), which gives the final solution \(y = \frac{b}{3a}\).

These manipulation steps involve the consistent application of mathematical operations, ensuring the balance and equality of equations. Mastering algebraic manipulation allows us to tackle more complex equations and ultimately find the values of unknown variables with ease.