Problem 30 - Algebra 1

Question

The variables x and y vary inversely. Use the given values to write an equation that relates x and y. $$x=1.5, y=50$$

Step-by-Step Solution

Answer

The equation that relates x and y is \(y = \frac{75}{x}\)

Step 1: Substitute Given Values

Plug the values of x and y given in the exercise into the formula \(y = \frac{k}{x}\). It transforms into \(50 = \frac{k}{1.5}\).

Step 2: Solve for k

Solving the equation for k gives \(k = 50 \times 1.5\), which equals 75.

Step 3: Write the Relationship

Now that we have the constant k, we can write the general relationship between x and y as \(y = \frac{75}{x}\)

Key Concepts

Algebraic Equations

Algebraic equations are mathematical statements that express the equality of two algebraic expressions. They consist of variables, constants, and operational symbols. When it comes to understanding inverse variation within algebraic equations, it's crucial to identify the inverse relationship between the two variables involved.

In the provided example, with given values of x and y, an equation relating the variables is written. The steps begin by substituting known values into a generalized equation and proceed to solving for the unknown constant. Mastery of algebraic equations not only comes from performing arithmetic operations but also from understanding how to manipulate equations to isolate and solve for desired variables.

Proportional Relationships

Proportional relationships are a foundational aspect of algebra where two quantities maintain a constant ratio. Direct variation is a specific type of proportional relationship where an increase in one quantity causes a proportional increase in the other.

Inverse variation, as presented in our exercise, is the opposite; as one variable increases, the other decreases at a rate that maintains a constant product, the constant of variation, k. Recognizing the difference between these types of variation can greatly enhance a student's ability to tackle related algebra problems.

Solving for Constants

In mathematics, constants are fixed values that don't change. Solving for a constant in an equation is often a step towards understanding the relationship between variables.

The given problem walks through finding the constant of variation, k, which is pivotal in defining the inverse relationship. By multiplying the given y-value with its corresponding x-value, as shown in 'Step 2', the constant is identified. This practice underscores the importance of constants in forming complete mathematical relationships and models.

Direct and Inverse Variation

Direct and inverse variation describe two distinct but related types of relationships between variables. Direct variation implies that as one variable increases, so does the other, at a consistent rate, defined by the equation y = kx, where k is the constant of variation.

In contrast, inverse variation, highlighted in the exercise, suggests a reciprocal relationship, represented by the equation \(y = \frac{k}{x}\). As x increases, y decreases such that their product, the constant k, remains the same. Understanding these contrasting relationships allows for the accurate modeling of real-world scenarios where such variations occur.