Problem 25 - Algebra 1

Question

Rewrite the equation so that \(y\) is a function of \(x .\) $$\frac{1}{3}(y+2)+3 x=7 x$$

Step-by-Step Solution

Answer

The given equation, when rewritten as a function of 'x', will be \(y = 12x - 2\).

Step 1: Simplification

This step involves simplification of the given equation. For this equation particular, we distribute the fraction \(\frac{1}{3}\) across \(y + 2\). This simplifies to \(\frac{1}{3}y + \frac{2}{3} + 3x = 7x\). Next, we can combine like terms by subtracting \(3x\) from both sides. This gives us \(\frac{1}{3}y + \frac{2}{3} = 4x\).

Step 2: Isolate 'y'

In this step, we aim to have 'y' on one side of the equation alone. We can do this by subtracting \(\frac{2}{3}\) from both sides. This will give us \(\frac{1}{3}y = 4x - \frac{2}{3}\).

Step 3: Solve for 'y'

To make 'y' the subject of the formula, we multiply through by 3. This results in \(y = 12x - 2\).

Key Concepts

Functions

In mathematics, functions are fundamental building blocks used to describe relationships between different quantities. A function, typically denoted as \( f(x) \), maps an input \( x \) to a single output, often represented as \( y \). The process of representing an equation where \( y \) becomes a function of \( x \), as seen in our exercise, is essential for understanding how one variable changes in response to another. Unlike equations that may have multiple solutions, a function provides a unique output for each input.

• Expressing \( y \) in terms of \( x \) helps us to understand how changes in \( x \) impact \( y \).
• Functions can be linear, quadratic, polynomial, and more—each having its own characteristic behavior.

Understanding functions enables us to model real-world scenarios, predict outcomes, and analyze relationships systematically. They serve as a vital tool in various fields, including economics, engineering, and statistics.

Equation Simplification

Equation simplification is the process of reducing an equation to its simplest form, making it easier to work with. In our example, simplification involved several key steps such as distributing common factors and combining like terms.

• By distributing \( \frac{1}{3} \) over \( y + 2 \), we broke the initial equation down into simpler components.
• Like terms, such as \( 3x \), were combined to streamline the equation further.

Simplifying the equation removes extraneous complexity, allowing easier manipulation and understanding. This step is often necessary before isolating variables or analyzing the equation, as it lays the groundwork by making the equation more manageable.
Whether solving algebraic equations or working with complex formulas, simplification is a pivotal skill that enhances clarity and efficiency.

Isolating Variables

Isolating variables involves manipulating an equation so that only one variable is on one side of the equation, leaving the rest on the other. This is a crucial step when expressing one variable as a function of another. In the given exercise, we needed to isolate \( y \) from the equation, putting it solely in terms of \( x \).

• Initially, \( \frac{2}{3} \) was subtracted from both sides, one step toward isolating \( y \).
• Finally, multiplying through by 3 achieved the complete isolation, resulting in \( y = 12x - 2 \).

Isolating variables is a fundamental algebraic technique enabling calculus operations, graph plotting, and further analysis. It translates real-world relationships into mathematical expressions that are easier to interpret and apply. Mastering this skill allows students to tackle a wider array of problems in physics, engineering, and economics with confidence.