Problem 17 - Algebra 1

Question

The variables x and y vary directly. Use the given values to write an equation that relates x and y. $$x=27, y=3$$

Step-by-Step Solution

Answer

The equation that relates x and y is \(y = \frac{1}{9}x\).

Step 1: Understanding Direct Variation

In direct variation, the two variables are directly proportional to each other. Therefore, we apply the formula for direct variation, which is \(y = kx\), where \(k\) is the constant of proportionality.

Step 2: Substituting Given Values

Next, we substitute the given values of \(x\) and \(y\) into the formula. We have \(x = 27\) and \(y = 3\), so we substitute these values to get \(3 = k * 27\).

Step 3: Solving for the constant of variation, \(k\)

To isolate \(k\), we divide both sides of the equation by 27. This gives us \(k = \frac{3}{27}\)

Step 4: Simplifying the value of \(k\)

Simplifying the fraction \(\frac{3}{27}\), we get \(k = \frac{1}{9}\).

Step 5: Writing the Equation

Finally we substitute the value of \(k\) we've found into the original equation of a direct variation to get our relation: \(y = \frac{1}{9}x\). This is the equation that relates \(x\) and \(y\).

Key Concepts

Constant of Proportionality

The constant of proportionality is a key concept in understanding direct variation. In a relationship where two variables are directly proportional, this constant, often denoted by the symbol \(k\), helps you understand how one variable changes as the other does. Imagine it as the fixed multiplier that connects the variables in a proportional relationship. In the example exercise, the constant of proportionality is derived when we know specific values for both \(x\) and \(y\). By substituting these known values into the equation \(y = kx\), we can solve for \(k\). Here, substituting \(x = 27\) and \(y = 3\), the equation becomes \(3 = 27k\). Solving for \(k\), we divide both sides by 27 to find \(k = \frac{1}{9}\). Understanding this constant helps you make predictions about how one variable affects another in real-world scenarios.

• It characterizes how two quantities are related.
• It remains constant as long as the conditions of direct variation are unchanged.
• It simplifies the mathematical model into a straightforward linear relationship.

Proportional Relationships

A proportional relationship exists when two variables maintain a consistent ratio. This relationship is defined by an equation where one variable is equal to a constant multiplied by the other, such as \(y = kx\). **In practical terms:** imagine two elements always moving in tandem. If you double one, the other doubles too. In the given exercise example, \(x\) and \(y\) have a direct variation, demonstrated by the equation \(3 = 27k\). By applying algebraic principles, we found \(k = \frac{1}{9}\), thus leading to the equation \(y = \frac{1}{9}x\). This shows clearly that when \(x\) increases or decreases, \(y\) will increase or decrease by a factor of \(\frac{1}{9}\).
*Recognizing a proportional relationship in equations:*

• Look for constancy in rates of change.
• Find the mathematical expression \(y = kx\), indicating a linear relationship.
• Consider practical examples like speed or density, where the relationship holds true.

Solving Equations

Solving equations is the process of finding the unknown values that satisfy the equation. When dealing with direct variation, like in our scenario, solving for the constant of proportionality \(k\) involves straightforward algebraic steps. Here's how it works: once we have the equation from the direct variation formula, substitute known quantities. For our example, we substitute \(x = 27\) and \(y = 3\) into the equation \(y = kx\). This substitution gives us the new equation \(3 = 27k\).
To solve this:

• Isolate \(k\) by rearranging the equation, which means divide both sides by the coefficient of \(k\), here 27.
• Calculate \(k = \frac{3}{27}\), which simplifies to \(k = \frac{1}{9}\).
• Replace \(k\) in the original equation to find the relationship \(y = \frac{1}{9}x\).

Solving equations helps cement the relationship between variables, ensuring you fully understand how one influences the other. These skills are fundamental in algebra and essential for tackling many real-world problems.