Problem 54 - Algebra 1

Question

The variables \(x\) and \(y\) vary directly. Use the given values of the variables to write an equation that relates \(x\) and \(y .\) $$x=33, y=9$$

Step-by-Step Solution

Answer

The equation that relates the variables \(x\) and \(y\) is \(y = 0.2727x\).

Step 1: Understand the Concept of Direct Variation

A direct variation is a relationship between two variables in which the ratio of the variables remains constant. This can be represented by the equation \(y=kx\), where the constant \(k\) is the constant of variation.

Step 2: Substitute the Given Values

Substitute the given values into the equation of variation. The values given are \(x=33\) and \(y=9\). So, the equation will look like this: \(9=k*33\).

Step 3: Solve for the Constant of Variation

To solve for \(k\), divide both sides of the equation by 33. That gives us \(k=9/33 = 0.2727\).

Step 4: Write the Final Equation

Now that we know the value of the constant of variation, we can write the final equation. Replace \(k\) in the original equation with the found value, we get \(y = 0.2727x\).

Key Concepts

Constant of Variation

The concept of the 'constant of variation' is pivotal in understanding direct variation. When two variables, such as x and y, vary directly, it’s like a promise that their relationship will always be consistent, dependable. This commitment is represented mathematically by a constant, denoted as k. It defines how changes in one variable are echoed by changes in the other.

In essence, the constant of variation is the ratio of y to x and it remains unchanged throughout the relationship. Think of it as the strict rule they follow: if x doubles, so does y, all thanks to that unwavering multiplier, k. For example, if we're given x = 33 and y = 9, we can calculate k by dividing y by x to get k = 9/33 = 0.2727. Now, no matter what new values of x we encounter, multiplying them by 0.2727 will give us the corresponding value of y.

Directly Proportional Relationships

In mathematics, we often talk about 'directly proportional relationships,' a fancy term that’s really quite simple at heart. It's the kind of bond between two things where one goes up or down, and so does the other, in perfect sync, like dance partners. In algebra, we express this relationship using the formula y = kx, showing how variable y is directly proportional to variable x through the constant of variation k.

What’s beautiful about these relationships is their simplicity and predictability. Increase x, and y will increase at a rate determined by k; decrease x, and y follows suit. This proportionality is a cornerstone concept not just in algebra, but it extends into various fields such as physics, economics, and biology, describing the steady rhythms of relationships in the natural and theoretical worlds alike.

Algebraic Equations

Algebraic equations are like the sentences of math; they tell us stories about numbers and the relationships between them. These stories are told using symbols like x and y, which represent unknown values that we seek to uncover. In the case of direct variation, the story is about how these variables move together, always keeping their ratio locked in place by the constant k.

To find the hidden value of k, we read the clues given: x = 33 and y = 9. We then substitute these clues into the equation y = kx to reveal the truth: k = 9/33. By solving this algebraic equation, we're not just finding a number; we’re uncovering the very essence of the relationship between x and y—the rule they live by. From there, we can rewrite the equation as y = 0.2727x, making it a useful tool to predict y for any x, and this rule of prediction is a powerful ally in a mathematician's toolkit.