Problem 62 - Algebra and Trigonometry

Question

Determine whether the equation defines y as a function of x. (See Example 10.) $$ 3 x+7 y=21 $$

Step-by-Step Solution

Answer

Yes, \( y = 3 - \frac{3}{7}x \) defines \( y \) as a function of \( x \).

Step 1: Rewrite the Equation

Start by rewriting the given equation in the form of a standard line equation where the dependent variable, \( y \), is explicitly expressed in terms of the independent variable, \( x \). The original equation is: \[ 3x + 7y = 21 \] To express \( y \) in terms of \( x \), isolate \( y \) on one side of the equation.

Step 2: Solve for y

Move the \( 3x \) term to the other side by subtracting it from both sides of the equation: \[ 7y = 21 - 3x \]Now, divide each term by 7 to solve for \( y \): \[ y = \frac{21 - 3x}{7} \] Simplify the equation further: \[ y = 3 - \frac{3}{7}x \]

Step 3: Analyze the Result

Now that we have \( y \) in terms of \( x \), check if each \( x \) value will yield a unique \( y \) value. In the equation \( y = 3 - \frac{3}{7}x \), for any given \( x \) value, there is only one corresponding \( y \) value. This confirms that the equation does define \( y \) as a function of \( x \).

Key Concepts

Linear Equation

A linear equation is a type of equation that forms a straight line when it is graphed on a coordinate plane. It typically follows the format: \[ ax + by = c \] where \( a \), \( b \), and \( c \) are constants. In this format, both the variables are first-degree, which means their exponents are 1.
This simplicity gives linear equations some unique properties, making them straightforward to solve and predictable in their behavior.

• They have either one solution, no solution, or infinitely many solutions.
• They can be rearranged into different forms, such as the slope-intercept form \( y = mx + b \), which directly shows the slope \( m \) and y-intercept \( b \).
• They can easily be graphed as straight lines, which makes visual interpretation simple.

For the given problem, rearranging the equation \( 3x + 7y = 21 \) into \( y = 3 - \frac{3}{7}x \) puts it into a familiar linear form that is easy to interpret.This linear form helps us determine whether \( y \) is a function of \( x \).

Dependent Variable

In the context of a function, the dependent variable is the output that depends on the input values of the independent variable. In this case, the dependent variable is \( y \), as it changes in response to different values of \( x \).
The dependent variable is typically isolated on one side of the equation when analyzing functions, as seen in our rearranged equation \( y = 3 - \frac{3}{7}x \).

• The value of \( y \) is determined once the value of \( x \) is known.
• In graphical terms, the dependent variable usually corresponds to the vertical axis (y-axis).
• Understanding the dependent variable helps analyze how outputs change as inputs vary.

By looking at our equation, any change in \( x \) directly affects \( y \), underlining the dependent nature of \( y \).

Independent Variable

The independent variable in an equation acts as the input that you have control over and which influences the dependent variable. For our function \( y = 3 - \frac{3}{7}x \), \( x \) is the independent variable. It is the variable we change freely to determine how \( y \) responds.
In the world of functions, the independent variable is often plotted on the horizontal axis of a graph (x-axis).

• It directly influences the dependent variable, affecting the output values.
• The equation's graph shape or direction can hint at how sensitive \( y \) is to changes in \( x \).
• Choosing different values for \( x \) shows how the function behaves and how \( y \) depends on \( x \).

Thus, understanding the role of \( x \) helps us comprehend the cause-effect relationship in a function.

Unique Solution

Unique solutions are quite crucial in mathematics, especially when analyzing functions. A unique solution in the context of a linear equation like \( 3x + 7y = 21 \) means that for each value of \( x \) in the equation \( y = 3 - \frac{3}{7}x \), there exists exactly one corresponding value of \( y \).
This characteristic is fundamental to confirming whether \( y \) is indeed a function of \( x \).

• A function must only produce one output for each input to be well-defined.
• This exclusivity ensures the function is clear and unambiguous.
• The concept of having one output per input fits into the vertical line test used in graphs to confirm a function's validity.

Here, our rearranged linear equation ensures that every value of \( x \) will lead to a unique solution for \( y \), verifying that \( y \) is a function of \( x \).