Problem 56 - 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 \(y .\) $$x=6.3, y=1.5$$

Step-by-Step Solution

Answer

The final equation of direct variation that relates \(x\) and \(y\) is \(y = 0.23809x\).

Step 1: Understanding Direct Variation

In direct variation, the variables \(x\) and \(y\) are related by a constant of variation, noted by \(k\), such that \(y=kx\). We need to find the value of \(k\) to write an equation that relates \(x\) and \(y\).

Step 2: Calculate 'k', the Variation Constant

Given \(x=6.3\) and \(y=1.5\), we substitute these values into the formula \(y=kx\) to find \(k\). Solving for \(k\) gives \(k= \frac{y}{x} = \frac{1.5}{6.3}\).

Step 3: Final Equation of Direct Variation

Calculate the value of \(k\) from the previous step and substitute it back into the equation \(y=kx\) to get the equation of direct variation relating \(x\) and \(y\).

Key Concepts

Variation Constant

In direct variation, understanding the variation constant is key to linking two variables. Let's clarify this with the concept of the variation constant, often symbolized by the letter \(k\). The variation constant \(k\) serves as a bridge between the variables \(x\) and \(y\) in a direct variation scenario. Here is how it works:

• Whenever \(y\) varies directly with \(x\), we mean \(y = kx\).
• The constant \(k\) remains unchanged and reflects how much \(y\) changes with every unit increase in \(x\).
• To find \(k\), we divide \(y\) by \(x\), as shown: \(k = \frac{y}{x}\).

In the given exercise, substituting the values \(y = 1.5\) and \(x = 6.3\) into the equation gives us \(k = \frac{1.5}{6.3}\). This fraction provides the variation constant as a specific number, representing the direct relationship's strength and nature.

Proportional Relationship

The idea of a proportional relationship is at the heart of direct variation. When two variables have a proportional relationship, their ratio, defined by the variation constant \(k\), remains consistent. This type of relationship is represented as:

• The variables increase or decrease together at a constant rate.
• The graph of a proportional relationship is a straight line through the origin (0,0).
• This relationship ensures consistency; changing one variable by a certain factor changes the other by the same factor.

In our previous example, we see that as \(x\) changes, \(y\) adjusts proportionally due to their shared constant of variation \(k\). If \(x\) doubles, \(y\) doubles too. This ties back to the basic property of direct variation: the consistency in ratio confirmed by the constant \(k\).

Linear Equation

When discussing direct variation, it naturally leads us to linear equations. In this type of equation, the relationship between \(x\) and \(y\) is captured in a linear form \(y = kx\). Let's break it down:

• A linear equation, in the context of direct variation, means \(y\) changes at a constant rate as \(x\) changes.
• The graph of \(y = kx\) is a straight line passing through the origin, making it a special case of linear equations called a direct proportionality line.
• The slope of this line is \(k\), symbolizing the rate at which \(y\) changes per unit change in \(x\).

In essence, understanding that \(y = kx\) is a linear equation allows one to visualize and interpret the proportional relationship graphically as a straight line, which is both simple and informative. In the exercise, once you find \(k\), you can construct and graph the equation to showcase how \(x\) and \(y\) relate dynamically.