Problem 27 - Algebra Form and Function

Question

Solve for the indicated variable. Assume all constants are non-zero. $$ y=3 \pi t, \text { for } t $$

Step-by-Step Solution

Answer

Question: Given the equation y = 3πt, solve for the variable t in terms of y. Answer: t = (y/3π)

Step 1: Write down the equation in the given form

We have the equation: $$ y = 3 \pi t $$ We want to solve this equation for t.

Step 2: Solving for t

To solve for t, we'll divide both sides of the equation by 3π: $$ \frac{y}{3 \pi} = \frac{3 \pi t}{3 \pi} $$ Now, the 3π terms on the right side of the equation cancel out: $$ \frac{y}{3 \pi} = t $$

Step 3: Write the final solution

Thus, the solution for t in terms of y is: $$ t = \frac{y}{3 \pi} $$

Key Concepts

Algebraic Manipulation

Algebraic manipulation is a fundamental skill in mathematics that enables you to solve equations and rearrange expressions to make them simpler or to isolate specific terms. In our example, we want to transform an equation to solve for a particular variable, here being the variable \( t \).

Here's how it works:

• Rearrangement: To isolate \( t \) in the equation \( y = 3 \pi t \), we rearrange the equation by performing the same operation on both sides.
• Inverse Operations: We use division, which is the inverse of multiplication, to solve for \( t \). We divide both sides by \( 3 \pi \) to maintain equality.

These manipulations help us get from an original equation to a desired format, like finding \( t \) in terms of \( y \). Algebraic manipulation is crucial for solving equations, as it provides the framework to systematically approach complex problems. By practicing these steps, you'll become more adept at quickly spotting the right manipulations needed in a given situation.

Variables and Constants

In mathematics, understanding variables and constants is pivotal. They're the backbone of equations and algebraic expressions.

• Variables: In our exercise, \( t \) and \( y \) are variables. They represent numbers that can change or take on different values.
• Constants: A constant, like \( 3 \pi \), is a fixed number that does not change. In equations, constants serve as known values that balance and define the relationship between variables.

Variables are marked by letters or symbols, often portraying quantities to be determined. Constants, on the other hand, are numbers like \( \,3\, ext{or} \, \pi\, \) (pi being approximately 3.14) that remain steady. This distinction is critical since solving equations often involves determining the value of a variable while constants help provide structure to the equation.

Solution Techniques

Solution techniques in algebra are strategies utilized to solve equations. These methods are essential for finding the value of our variables. Consider solving \( y = 3 \pi t \) for \( t \):

• Isolating the Variable: The goal is to have the variable \( t \) by itself on one side of the equation. It often involves operations such as addition, subtraction, multiplication, or division.
• Balancing the Equation: To maintain equality, whatever operation you perform on one side of the equation, you must do the same on the other. This technique ensures the equation remains balanced throughout your work.
• Checking the Solution: Once you've isolated \( t \), inserting your solution back into the original equation verifies its correctness.

By applying these techniques, we've shown that \( t = \frac{y}{3\pi} \). While this was a straightforward example, more complex equations will require a combination of these different techniques. Solving equations is much like solving a puzzle: piecing together operations in a logical sequence to uncover the desired solution.