Problem 14 - 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 \(y .\) $$ x=18, y=6 $$

Step-by-Step Solution

Answer

The equation that relates \(x\) and \(y\) in the case of direct variation, using the given values, is \(x = 3y\).

Step 1: Understanding the Concept of Direct Variation

If two variables \(x\) and \(y\) vary directly, it means they maintain a constant ratio. In other words, there is a constant \(k\) such that \(x/y = k\). This can also be written as \(y = (1/k)x\) or \(x = ky\). The goal in this exercise is to find \(k\), the constant of variation.

Step 2: Calculate the Constant of Variation

To find the constant of variation \(k\), we can use the given values of \(x\) and \(y\), which are 18 and 6 respectively. Substituting these into the equation for direct variation: \(k = x / y = 18 / 6 = 3\).

Step 3: Write the Equation that Relates x and y

Now, with the constant of variation \(k = 3\), we can write the equation that represents the direct variation between \(x\) and \(y\). Therefore, \(x = ky\) becomes \(x = 3y\).

Key Concepts

Variable Relationship in Algebra

In Algebra, understanding the relationships between variables is critical for solving equations and modeling real-world scenarios. Variable relationships can take many forms, such as direct variation, inverse variation, or more complex functions. When two variables are in direct variation, they increase or decrease in tandem. This means that as one variable goes up, the other one also goes up by a proportionate amount, which is expressed by a constant. It's like having two dancers moving together in perfect sync—one step forward for one means one step forward for the other. Here's how you can spot a direct variation:

• If you increase variable x, variable y increases as well.
• If you decrease variable x, variable y decreases too.
• The ratio of x to y always remains constant, meaning if you divide x by y, you always get the same number.

The concept becomes particularly handy when you need to predict the value of one variable given the other; this linear relationship simplifies many seemingly complex problems.

Constant of Variation

In the case of direct variation, the 'constant of variation', typically denoted by k, acts like a multiplier that outlines the relationship between two variables. To make it more relatable, consider k as a secret ingredient in a recipe that precisely determines how the attributes of the dish change based on the amount used. Similarly, in algebra, k helps us understand the precise nature of the change between the variables.

Whenever two variables are involved in direct variation, and we're given their specific values, finding k is as straightforward as dividing one variable by the other (usually y by x). Once we have k, it becomes the foundation to build an equation that models the relationship. For instance, knowing that k equals 3 when x is 18 and y is 6 gives us a powerful piece of information to predict either variable in different situations.

Writing Algebraic Equations

Algebraic equations are like puzzles, and writing them involves arranging the known and unknown pieces to create a clear picture. When dealing with direct variation, the equation typically takes the form y = kx or x = ky, with k being the constant of variation. Imagine this equation being a summary of a story that tells you exactly how the variables relate to each other.

Writing the correct algebraic equation is like drawing the right map. If you know where 'X' marks the spot, you can figure out how to get there from any other point. Let's say we uncover a constant of variation k of 3 for our variables x and y. The equation we write, x = 3y, then becomes a guide for understanding how any value of y can lead us to the corresponding value of x. It's essentially the formula for a secret code that, once known, can be used to unlock countless solutions.

Proportional Relationships

Proportional relationships are a cornerstone in algebra and represent a form of direct variation. When we say two quantities are proportional, it means they have a direct relationship that can be described by a constant ratio. This is similar to saying that two things are scaled versions of each other. Just like model cars are scaled-down versions of the real thing while maintaining the same proportions, proportional variables maintain a consistent ratio or scale.

• In the equation y = kx, a proportional increase in x will result in a proportional increase in y.
• This constant ratio k indicates how one variable changes in relation to another.

The beauty of proportional relationships lies in their predictability. By understanding the constant ratio, one can predict or calculate one variable based on the other, allowing for a relative ease in solving problems that might otherwise seem opaque. It's akin to having a universal converter between two different units of measurement, enabling efficient and precise conversions.