Problem 8 - 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. $$ y=\sqrt{5} \cdot x $$

Step-by-Step Solution

Answer

If so, what is the constant of proportionality? Answer: Yes, \(y\) is directly proportional to \(x\), and the constant of proportionality is \(\sqrt{5}\).

Step 1: Identify if y is directly proportional to x

First, let's recall the definition of direct proportionality: Two variables, \(x\) and \(y\) are directly proportional if there exists a nonzero constant \(k\) such that \(y = k \cdot x\). In our given formula, \(y = \sqrt{5} \cdot x\), we can observe that \(y\) depends only on \(x\) and there is a constant \(\sqrt{5}\) multiplying \(x\). Hence, \(y\) is directly proportional to \(x\).

Step 2: Find the constant of proportionality

Now that we have confirmed that \(y\) is directly proportional to \(x\), we can identify the constant of proportionality, \(k\). Based on the definition, \(k\) is the constant that relates \(y\) and \(x\) in the equation \(y = k \cdot x\). In our given formula, the constant of proportionality is \(\sqrt{5}\). So, the constant of proportionality is \(\boxed{\sqrt{5}}\).

Key Concepts

Constant of Proportionality

The constant of proportionality is a key concept in understanding the relationship between two directly proportional variables. When we say two variables are directly proportional, it means that if one variable changes, the other changes in a predictable manner. This relationship is defined by a nonzero constant, commonly represented as \( k \).

For instance, in the equation \( y = k \cdot x \), \( k \) is the constant of proportionality. It tells us how much \( y \) will change as \( x \) changes. If \( k \) were 2, for every increase of 1 in \( x \), \( y \) would increase by 2.

• The formula ensures proportionality as long as \( k eq 0 \).
• In the example \( y = \sqrt{5} \cdot x \), \( \sqrt{5} \) is the constant of proportionality.

Recognizing this constant helps us predict and calculate changes in the variables' values.

Mathematical Formulas

Mathematical formulas are expressions that use symbols and variables to denote relationships and rules in math. They are precise tools that help us describe real-world phenomena in a concise way. In the case of direct proportionality, the formula \( y = k \cdot x \) is simple but powerful.

This formula conveys a lot of information with minimal notation. It not only shows a connection between \( x \) and \( y \), but it also provides the means to calculate one if we know the other.

• The formula is useful across various fields such as physics, economics, and engineering.
• Proportional formulas help model real-life situations like speed, cost, and time relationships.

Employing such formulas allows us to apply mathematical reasoning to solve practical problems rigorously.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations. They form the basis of algebra, enabling us to generalize and solve equations. Understanding these expressions is crucial for simplifying and analyzing mathematical problems.

Direct proportionality involves algebraic expressions like \( y = \sqrt{5} \cdot x \). This expression indicates a direct relationship and simplifies our analysis.

• Variables like \( x \) and \( y \) represent unknown or changing quantities.
• Arithmetic operations, such as multiplication, link the variables with the constant of proportionality.

By interpreting algebraic expressions, we can gain insights into relationships and make informed predictions in diverse contexts.