Problem 15 - Contemporary Precalculus

Question

Determine whether the equation defines \(y\) as a function of \(x\) or defines \(x\) as a function of \(y\) $$3 x+2 y=12$$

Step-by-Step Solution

Answer

Answer: Yes, the equation can be expressed as both y being a function of x (y = -3/2x + 6) and x being a function of y (x = -2/3y + 4).

Step 1: Express y in terms of x

To check if y is a function of x, let's try to rewrite the equation in a way that isolates y. This will allow us to see if y can be expressed as a function of x. $$3x+2y=12$$ $$2y=-3x+12$$ $$y=-\frac{3}{2}x+6$$ This expression shows that y can be represented as a function of x.

Step 2: Express x in terms of y

Similarly, we can try to express x in terms of y to check if x can be as a function of y. $$3x+2y=12$$ $$3x=-2y+12$$ $$x=-\frac{2}{3}y+4$$ This expression also demonstrates that x can be represented as a function of y.

Step 3: Conclusion

Through the process of isolating y and x in the given equation, we found that both scenarios are possible. The equation can represent y as a function of x (y = -\frac{3}{2}x+6) as well as x as a function of y (x = -\frac{2}{3}y+4).

Key Concepts

Relations

When exploring mathematical relations, it's essential to understand the bond between two entities, such as variables. A relation in mathematics is any set of ordered pairs, where each pair connects particular elements from two sets. In the exercise provided, we have the equation \(3x + 2y = 12\), which forms a relation between \(x\) and \(y\). This relation indicates how \(x\) and \(y\) are linked through the given equation.

In this specific relation:

• \(x\) is known as the independent variable because we can choose its values freely.
• \(y\) is generally the dependent variable because its value depends on \(x\).
• Both variables can define one another under certain conditions, which is explored in the solution step-by-step.

Recognizing relations is the first step in determining whether one variable is a function of another, or vice versa. They are foundational to understanding real-world applications, from calculating interest to converting currency rates.

Linear Equations

Linear equations are straightforward but crucial concepts in algebra. They have a strong presence in many mathematical applications because they create straight lines when graphed on a coordinate plane. The equation \(3x + 2y = 12\) is a classic example of a linear equation with two variables. The terms in linear equations are either constants or products of a constant and a single variable.

Let's review the essential features of linear equations:

• Each equation forms a straight line, characterized by its slope and y-intercept when in slope-intercept form \(y = mx + b\).
• The linear equation \(3x + 2y = 12\) can be rewritten in this form to find the slope \(-\frac{3}{2}\) and y-intercept \(6\).
• This equation can be manipulated to isolate either \(x\) or \(y\), showing flexibility in uncovering the function relationship.

Understanding linear equations helps in predicting future values and depicting real-life situations involving constant rates of change, such as speed and work rates.

Expression of Variables

The expression of variables is a technique where we manipulate equations to "solve for" a particular variable. In this exercise, both \(x\) and \(y\) are expressed in terms of one another, demonstrating their functional relationships.

To express a variable:

• Identify the variable you want to isolate. For instance, when we express \(y\) in terms of \(x\), we rearrange the equation \(3x + 2y = 12\) to get \(y = -\frac{3}{2}x + 6\).
• This equation now explicitly defines \(y\) as a function of \(x\), showing how \(y\) changes with different \(x\).
• Similarly, expressing \(x\) as \(-\frac{2}{3}y + 4\) illustrates \(x\) as a function of \(y\).

Mastering the expression of variables by isolating them signifies a deep understanding of function relationships and is essential in solving various mathematical and real-world problems.