Problem 65 - College Algebra with Modeling and Visualization

Question

Solve the equation for \(y .\) Determine if y is a function of \(x\). $$ 4 x^{2}+3 y=\frac{y+1}{3} $$

Step-by-Step Solution

Answer

Yes, \(y\) is a function of \(x\).

Step 1: Eliminate the fraction

To eliminate the fraction, multiply every term in the equation by 3 to get rid of the denominator. This gives:\[ 3(4x^2 + 3y) = y + 1 \]Which simplifies to:\[ 12x^2 + 9y = y + 1 \]

Step 2: Solve for y

To isolate \(y\), first subtract \(y\) from both sides of the equation:\[ 12x^2 + 9y - y = 1 \]This simplifies to:\[ 12x^2 + 8y = 1 \]Now, isolate \(y\) by subtracting \(12x^2\) from both sides:\[ 8y = 1 - 12x^2 \]Finally, divide each term by 8 to solve for \(y\):\[ y = \frac{1 - 12x^2}{8} \]

Step 3: Determine if y is a function of x

The equation we found for \(y\) is \(y = \frac{1 - 12x^2}{8}\), which assigns a single value of \(y\) for every \(x\). In mathematical terms, \(y\) is dependent on \(x\) in a way that each \(x\) has exactly one associated \(y\). Thus, \(y\) is a function of \(x\).

Key Concepts

Functions

In mathematics, functions are like machines that take an input value and transform it into an output value. Precisely, a function assigns each input exactly one output. In this context, we have the equation \[ y = \frac{1 - 12x^2}{8} \]which represents \(y\) as a function of \(x\). Here’s how we know this:

• For any given \(x\), there is one, and only one, corresponding \(y\) value.
• All operations used in the equation ensure that as \(x\) changes, \(y\) changes in a predictable manner.

Thus, the equation adheres perfectly to the definition of a function. This relationship is crucial because it tells us that we can determine \(y\) for any given \(x\), ensuring no ambiguity in the result.

Fraction Elimination

Fractions can make equations complex and harder to solve, so we often eliminate them by removing the denominators. The aim is to simplify the equation to make further steps more manageable. In our equation\[ 4x^2 + 3y = \frac{y+1}{3} \]we have a fraction with a denominator of 3. To eliminate this fraction:

• Multiply every term in the equation by 3, which is the denominator of the fraction.
• This multiplication cancels out the denominator, leaving you with a simpler equation.\[ 3(4x^2 + 3y) = y + 1 \]Streamlining in this way is a common practice because it turns a cumbersome equation into a simpler, more direct form. This step paves the way for easier isolation and solving of variables.

Isolation of Variables

The goal of isolating a variable is to express it in terms of other variables or constants. We focus on having the equation written so that the variable of interest stands alone on one side of the equation. In our example, the equation\[ 12x^2 + 8y = 1 \]is neatly arranged to solve for \(y\). Here’s the process:

• First, simplify by bringing like terms together. Subtract \(y\) on both sides to simplify the equation.
• Continue by isolating \(y\). Since it is multiplied by 8, you divide every term by 8 to solve for \(y\):\[ y = \frac{1 - 12x^2}{8} \]

Through this careful process, we express \(y\) in terms of \(x\), clear of other terms and simplified. This is crucial for understanding relationships between variables in an equation.