Problem 50 - Algebra 1

Question

The variables x and y vary directly. Use the given values to write an equation that relates x and y. $$x=6, y=42$$

Step-by-Step Solution

Answer

The equation that relates the variables x and y is \(x = \frac{1}{7}y\).

Step 1: Understand direct variation concept

Given a direct variation between two variables, if x varies directly as y, this can be written as \(x=ky\), where k is the constant of variation.

Step 2: Substitute the given values

Given that x=6 and y=42, we can substitute these values into the equation to find the value of the constant k. So, \(6=k*42\).

Step 3: Solve for k

By dividing both sides of the equation by 42, we get \(k = \frac{6}{42}\).

Step 4: Simplify k

After simplifying the equation we get \(k = \frac{1}{7}\).

Step 5: Write the equation that relates x and y

By substituting the value of k back into the direct variation formula, we obtain the equation that relates the variables x and y, which is \(x = \frac{1}{7}y\).

Key Concepts

Constant of Variation

In direct variation, we often come across the term "constant of variation." This constant, denoted as \( k \), serves as the proportionality factor that links two variables that vary directly with each other. When understanding direct variation, think of it as a scaled relationship between two quantities.
In simpler terms, if you know one variable, the constant of variation helps you find the other. For the exercise above, our task is to determine the constant \( k \) using given values for \( x \) and \( y \).
We start with the general form:

• If \( x = ky \), it means \( x \) changes with \( y \), and \( k \) is what keeps them consistent.
• Given \( x = 6 \) and \( y = 42 \), we substitute to solve for \( k \).
• Set up the equation: \( 6 = k \times 42 \).
• Solving gives \( k = \frac{6}{42} \), which simplifies to \( \frac{1}{7} \).

So, \( \frac{1}{7} \) is our constant of variation for this problem.

Equation of Direct Variation

Once you've determined the constant of variation \( k \), you can formulate the equation that shows the direct relationship between the variables. This is known as the "equation of direct variation."
A direct variation equation allows us to express one variable in terms of the other. In this context, the equation takes the form:

• Starting with \( x = ky \), substitute the value of \( k \) that we found: \( k = \frac{1}{7} \).
• This gives us the relationship: \( x = \frac{1}{7}y \).

This equation tells us that whatever \( y \) is, \( x \) will be \( \frac{1}{7} \) times that value. This concise formula lets you predict \( x \) for any given \( y \). It's a powerful tool in problems involving direct variation, as it simplifies understanding how changes in one variable affect the other.

Solving Equations

Solving equations in the context of direct variation involves a straightforward method. The goal is to isolate and identity the constant of variation \( k \) and then use it to express the relationship between the variables.
In the exercise, the steps can be broken down into manageable parts:

• Start with the given values: \( x = 6 \) and \( y = 42 \).
• Use the direct variation formula \( x = ky \) to substitute these values into the equation, forming \( 6 = k \times 42 \).
• Divide both sides by 42 to solve for \( k \): \( k = \frac{6}{42} \).
• Simplify the fraction \( \frac{6}{42} \) to \( \frac{1}{7} \), which represents the constant.

This step-by-step method ensures that you accurately identify the constant and write the equation correctly. Solving such equations not only solidifies your understanding of direct variation but also enhances your overall problem-solving skills. It's important to approach each step methodically to avoid errors and set a solid foundation for more complex problems in algebra.