Problem 10 - Algebra 2

Question

Determine whether \(y\) varies directly with \(x .\) If so, find the constant of variation. \(y=6 x\)

Step-by-Step Solution

Answer

Yes, `y` varies directly with `x`, and the constant of variation is 6.

Step 1: Understanding Direct Variation

A direct variation can be identified if the equation has the form of `y = kx`, where `k` is the constant of variation. The variable `y` varies directly with `x` if `y` is equal to `x` times a constant.

Step 2: Comparing to the Direct Variation Form

The given equation is `y = 6x`. This equation matches the direct variation form `y = kx` where `k` is the coefficient of `x`.

Step 3: Identifying the Constant of Variation

From the equation `y = 6x`, the coefficient of `x` is 6. Therefore, `k = 6`. This is the constant of variation.

Key Concepts

Constant of Variation

When exploring the mysteries of algebra, one might encounter the term 'constant of variation'. This sounds like a grand concept, right? Well, it's an essential piece of the puzzle in understanding direct variation relationships.

Picture a scenario where one quantity depends on another in such a precise way that if you double one, the other doubles too. Think of it as an unchanging multiplier that connects two varying quantities, hence 'constant of variation'. Now, in our exercise, the equation given is \( y = 6x \), which is a classic direct variation example.

To pinpoint the constant of variation, simply look at the number snugly sitting next to the \( x \). Here, that number is 6. This tells us that for every unit increase in \( x \), \( y \) shoots up by 6 units. It's like a secret code between \( x \) and \( y \): whatever \( x \) is, \( y \) will be 6 times that!

Algebraic Equations

Algebraic equations are like treasure maps, leading us to the hidden jewels of unknown values. They consist of symbols and numbers woven together with mathematical operations, creating a balanced expression.

In the realm of direct variation, our algebraic equation takes the simple, yet elegant form of \( y = kx \). Dare to call it simplistic? It's actually quite versatile! This equation represents a straight line with a slope equivalent to our constant of variation, \( k \), when plotted on a graph. It's a marvellous tool that allows us to predict and understand how one thing changes in relation to another.

In our textbook problem, the equation \( y = 6x \) stands proudly, showcasing that it follows this elegant pattern. If \( y \) and \( x \) were dancers, \( 6 \) would be their rhythm, dictating how in sync their movements are!

Proportional Relationships

Let's dive into the world of proportional relationships, which is like finding harmony in numbers. When two quantities are serving couple goals by increasing or decreasing in sync, that's proportionality in action.

In the spotlight, direct variation is a special type of proportional relationship where one quantity (let's say \( y \)) is always a constant multiple (our friend, the constant of variation) of another quantity (hello, \( x \)). The beauty here is consistency; no matter what values \( x \) and \( y \) fancy taking on, they remain faithful to their proportional bond.

The equation \( y = 6x \) from our exercise is a testament to this steadfast relationship. As \( x \) pirouettes through different values, \( y \) follows, maintaining a graceful six-fold distance. It's this dependability that makes proportional relationships so reliable and, frankly, a cornerstone of algebraic concepts.