Problem 9 - Algebra Form and Function

Question

For each of the formulas in Exercises 5-13, is \(y\) directly proportional to \(x ?\) If so, give the constant of proportionality. $$ y=x / 9 $$

Step-by-Step Solution

Answer

If so, what is the constant of proportionality? Answer: Yes, y is directly proportional to x in the given formula. The constant of proportionality is \(k = \frac{1}{9}\).

Step 1: Compare the given formula to the direct proportionality form

We are given the formula \(y = \frac{x}{9}\). To compare this to the form \(y = kx\), we can rewrite the given formula as \(y = kx\) with \(k=\frac{1}{9}\).

Step 2: Determine if \(y\) is directly proportional to \(x\)

Since the given formula can be rewritten as \(y = kx\), with \(k=\frac{1}{9}\), it is clear that \(y\) is directly proportional to \(x\). The constant of proportionality is \(k=\frac{1}{9}\).

Step 3: State the conclusion

In the given formula \(y = \frac{x}{9}\), \(y\) is directly proportional to \(x\), and the constant of proportionality is \(k = \frac{1}{9}\).

Key Concepts

Constant of proportionality

In a direct proportionality scenario, the constant of proportionality (\(k\)) plays a central role. Essentially, it is a fixed number that relates two variables, such as \(y\) and \(x\) in the equation \(y = kx\). This constant tells us how much \(y\) changes when \(x\) changes. For example, in our exercise, the formula \(y = \frac{x}{9}\) can be rearranged to \(y = kx\) with \(k = \frac{1}{9}\). This means that for every 1 unit increase in \(x\), \(y\) increases by \(\frac{1}{9}\) units.

• The constant \(k\) will always be a number by which you multiply \(x\) to find \(y\)
• The value of \(k\) remains unchanged for any corresponding values of \(x\) and \(y\)
• If \(k\) is greater than 1, \(y\) increases faster than \(x\); if less, it increases slower

Understanding \(k\) helps determine the rate and direction of a change in proportional relationships.

Proportional relationships

Proportional relationships represent a special kind of linear relationship where two quantities change at a consistent rate relative to one another. In simple terms, \(y\) and \(x\) are directly linked. When there's a direct proportionality between these variables, they maintain the equation form \(y = kx\), where \(k\) is the constant of proportionality.

Such a relationship means:

• If \(x\) doubles, \(y\) also doubles.
• The graph of a proportional relationship is a straight line that passes through the origin (0,0).
• The slope of this line is the constant of proportionality, \(k\).

For example, in our formula \(y = \frac{x}{9}\), \(y\) changes at a steady rate of \(\frac{1}{9}\) as \(x\) changes. This stability and predictability characterize proportional relationships, making them easy to work with in both theoretical and practical applications.

Algebraic expressions

Algebraic expressions are a combination of numbers, variables, and operations that represent a specific mathematical situation or relationship. They offer a way to generalize and solve problems. In the exercise, the algebraic expression \(y = \frac{x}{9}\) serves to define a specific proportional relationship between \(y\) and \(x\).

Key points about algebraic expressions:

• They can consist of one or more terms. Here, it is a single-term expression.
• Variables in the expression, like \(x\) and \(y\), allow for wide-ranging calculations and flexibility.
• They help in forming equations and inequalities.

Using algebraic expressions helps in simplifying complex relationships and performing calculations. In the context of the exercise, the expression highlights the proportional link between two variables, with \(\frac{1}{9}\) acting as the proportionality constant. By manipulating algebraic expressions, you can explore various scenarios and solve a wide range of mathematical problems.