Problem 23 - Algebra and Trigonometry Real Mathematics, Real People

Question

Determine whether the equation represents \(y\) as a function of \(x.\) $$2 x+3 y=4$$

Step-by-Step Solution

Answer

Yes, the given equation represents \(y\) as a function of \(x\).

Step 1: Rearrange the equation

The equation needs to be solved for \(y\). Start by subtracting \(2x\) from both sides of the equation to isolate \(3y\). This results in \(3 y = 4 - 2x\).

Step 2: Isolate y

Next, divide every term by 3 to solve for \(y\). This results in the equation \(y = \frac{(4 - 2x)}{3}\)

Step 3: Determine whether y is a function of x

Now that \(y\) has been isolated, check whether \(y\) is a function of \(x\). In the equation \(y = \frac{(4 - 2x)}{3}\), for any given \(x\), there is exactly one corresponding value of \(y\). This means that the equation represents \(y\) as a function of \(x\).

Key Concepts

Equations

An equation is a mathematical statement that asserts the equality of two expressions. Equations can involve numbers, variables, or a combination of both. They often contain an equal sign to represent this relationship. There are different types of equations, such as linear, quadratic, and exponential equations. In this exercise, we are dealing with a linear equation, which generally takes the form: \[ Ax + By = C \] where \(A\), \(B\), and \(C\) are constants. Understanding equations is vital because they form the basis of many mathematical disciplines. Solving equations often involves finding the values of the variables that satisfy the equation, known as solutions. Equations can represent various real-world scenarios, from calculating interest to predicting population growth.

Solving for a variable

Solving for a variable is the process of manipulating an equation to express one variable in terms of the others. This usually involves performing a series of operations that include addition, subtraction, multiplication, or division on both sides of the equation to isolate the desired variable. In our exercise, we start with the equation:\[2x + 3y = 4\] To solve for \(y\), follow these steps:

• Subtract \(2x\) from both sides to isolate the term containing \(y\):

\[3y = 4 - 2x\]

• Divide every term by 3 to completely isolate \(y\):

\[y = \frac{(4 - 2x)}{3}\] The result is an expression of \(y\) in terms of \(x\), which is crucial for further analysis like determining if \(y\) is a function of \(x\). These steps highlight the importance of understanding operations and their properties when working with equations.

Linear functions

A linear function is a function that creates a straight line when graphed. These functions are represented by equations of the form:\[y = mx + b\]where \(m\) is the slope and \(b\) is the y-intercept. Linear functions are characterized by the fact that they have a constant rate of change.In our exercise, after solving for \(y\), we get the equation:\[y = \frac{4}{3} - \frac{2}{3}x\]This is a linear function in the form \(y = mx + b\), where the slope \(m\) is \(-\frac{2}{3}\) and the y-intercept \(b\) is \(\frac{4}{3}\).Linear functions are simple yet powerful in modeling situations where there is a consistent relationship between variables. For example, if you are paid hourly, your pay can be modeled by a linear function because you earn the same amount per hour worked. Understanding linear functions allows one to analyze and interpret these kinds of relationships effectively.