Problem 23 - Precalculus with Limits

Question

In Exercises 19-36, determine whether the equation represents \(y\) as a function of \(x\). \(2x + 3y = 4\)

Step-by-Step Solution

Answer

Yes, the equation \(2x + 3y = 4\) represents \(y\) as a function of \(x\). After rearranging, we find \(y = (4 - 2x) / 3\). For any value of \(x\), there is only one corresponding value of \(y\).

Step 1: Arrange the equation into slope-intercept form

Start with the original equation \(2x + 3y = 4\). The goal is to isolate \(y\) on one side of the equation. First, subtract \(2x\) from both sides to move it to the other side of the equation. This gives \(3y = 4 - 2x\).

Step 2: Isolate y

Now, divide every term by 3 to solve for \(y\), giving \(y = (4 - 2x) / 3\).

Step 3: Confirm y as a function of x

Now the equation is in the form \(y = mx + b\). For every value of \(x\), there is only one corresponding value of \(y\), meaning \(y\) is a function of \(x\).

Key Concepts

Slope-Intercept Form

Understanding the slope-intercept form is crucial for analyzing and graphing linear equations. It is expressed as \(y = mx + b\), where \(m\) represents the slope of the line and \(b\) indicates the y-intercept, which is the point where the line crosses the y-axis. In the context of the given exercise, the equation \(2x + 3y = 4\) was rearranged into slope-intercept form by isolating \(y\).

This manipulation, shown in the step-by-step solution, simplifies the identification of the slope and y-intercept. For instance, once the equation is in the form \(y = \frac{4 - 2x}{3}\), it's clear that the slope (\(m\)) is \(-\frac{2}{3}\) and the y-intercept (\(b\)) is \(\frac{4}{3}\). These constants allow us to quickly sketch the graph of the equation and understand how \(y\) changes with each increment of \(x\). Slope indicates the steepness of the line and the direction of change, while the intercept provides a starting point for drawing the line on a graph.

Isolating y in Equations

Isolating \(y\) in an equation is a fundamental algebraic skill that helps in identifying functions and graphing linear equations. To isolate \(y\), you must manipulate the equation so that \(y\) stands alone on one side of the equality, with all other terms on the opposite side.

In the given problem, the process began with the original equation \(2x + 3y = 4\). The equation was reorganized by subtracting \(2x\) from both sides, leading to \(3y = 4 - 2x\). Afterwards, dividing each term by 3 resulted in \(y = \frac{4 - 2x}{3}\), successfully isolating \(y\). It’s important to ensure all operations are balanced on each side of the equation and to simplify the terms when possible. This approach not only reveals the function nature of the relationship between \(x\) and \(y\) but also prepares the equation for graphing or further analysis.

Identifying Functions

To determine whether an equation represents \(y\) as a function of \(x\), each input for \(x\) must correspond to exactly one output for \(y\). In other words, for every value of \(x\), there can only be one associated value of \(y\).

Using the slope-intercept form makes identifying functions more straightforward. As seen in the solved exercise, after isolating \(y\) to obtain \(y = \frac{4 - 2x}{3}\), it was observed that for any given value of \(x\), there is a single resulting value of \(y\), hence satisfying the definition of a function. This one-to-one relationship is the hallmark of functional dependence between variables, which is fundamental in mathematics for describing consistent relationships and for establishing the basis for calculus concepts such as limits and continuity.