Problem 17 - Algebra 1: Concepts and Skills

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 \(x = 9y\)

Step 1: Understand the relationship

Given that \(x\) and \(y\) are directly proportional, meaning the ratio of \(x\) to \(y\) is a constant. This is mathematically represented as \(x=k*y\) where \(k\) is the constant of variation.

Step 2: Solve for the constant of variation

To find the constant of variation \(k\), substitute the given values of \(x=27\) and \(y=3\) into the proportionality equation \(x=k*y\). Solving for \(k\) gives \(k=x/y = 27/3 = 9\).

Step 3: Write the equation that relates x and y

We substitute \(k=9\) into \(x=k*y\) to obtain our equation \(x=9*y\). This is the required equation that relates x and y for all such directly proportional sets.

Key Concepts

Proportionality

Proportionality is a mathematical concept describing a relationship where two quantities increase or decrease at the same rate. When two variables are directly proportional, as in the relationship between variables \(x\) and \(y\) given in this exercise, the ratio of the two quantities remains constant. This means if one variable is doubled, the other will double as well, and if one is halved, the other also halves.

Understanding this relationship is key because it allows us to express the interaction between \(x\) and \(y\) in a simplified form: \(x = k \cdot y\). Here, \(k\) is the constant of variation, which we will discuss further. Direct proportionality can be visualized as a straight line through the origin on a graph with \(k\) being the slope of that line.

To recognize a directly proportional relationship in word problems or equations, look for phrases like "constant rate," "constant ratio," or "directly varies." Identifying these indicators will help set up and solve proportionality equations effectively.

Constant of Variation

In a directly proportional relationship, the constant of variation, often denoted as \(k\), is the factor that defines the specific link between the two variables. It is the unchanging number that indicates how one variable scales with another. For the given exercise, finding \(k\) involves dividing the value of \(x\) by \(y\).

Let's break this down further:

• Given data: \(x = 27\) and \(y = 3\).
• Using the formula for direct variation: \(x = k \cdot y\), substitute the known values.
• Solve for \(k\) by rearranging: \(k = \frac{x}{y} = \frac{27}{3} = 9\).

This calculation reveals that for every time \(x\) increases or decreases, \(y\) changes by a value determined by \(k = 9\). This constant provides a necessary link to formulate the final equation that represents the direct proportionality condition.

Linear Equation

A linear equation expresses the relationship between two variables in a straight-line graph. In the context of direct variation, we see this linear equation formulated as \(x = k \cdot y\). This means the equation is simple, straightforward, and easy to graph or manipulate mathematically.

For our specific problem, after determining that \(k = 9\), the linear equation becomes \(x = 9 \cdot y\). This equation tells us that \(x\) is always nine times the size of \(y\) in its value. Implementing and understanding this equation helps to find corresponding values for \(x\) or \(y\) when one is known, maintaining the constant ratio set by direct proportionality.

With linear equations in direct variation, remember they always pass through the origin (the point \((0, 0)\)), as when \(y = 0\), \(x\) is also \(0\). Recognizing this property can help verify if your solution is correctly representing a true directly proportional relationship.