Problem 54 - Algebra 1

Question

The variables x and y vary directly. Use the given values to write an equation that relates x and y. $$x=\frac{3}{4}, y=3$$

Step-by-Step Solution

Answer

The equation relating \(x\) and \(y\) in a direct variation is \(y = 4x\).

Step 1: Understand Direct Variation

In problems of direct variation, the variables \(x\) and \(y\) are connected by a constant of variation. This is expressed as \(y=kx\). In this problem, \(x = \frac{3}{4}\) and \(y = 3\). We can use these values to find the constant of variation \(k\).

Step 2: Solve for the Constant of Variation

Insert the given values of \(x\) and \(y\) into the equation to solve for \(k\). So we get: \(3 = k * \frac{3}{4}\). To isolate \(k\), divide both sides of the equation by \(\frac{3}{4}\). This gives: \(k = 4\).

Step 3: Writing the Direct Variation Equation

Now that the constant of variation \(k\) is known, the general direct variation equation can be written as \(y = kx\) or \(y = 4x\).

Key Concepts

Constant of Variation

In direct variation, the constant of variation, denoted by \(k\), acts as the bridge connecting two variables \(x\) and \(y\). They have a linear relationship represented by the equation \(y = kx\). The constant \(k\) keeps the relationship consistent. Once you find \(k\), you determine how much \(y\) changes as \(x\) changes.
Looking at our problem: we have the values \(x = \frac{3}{4}\) and \(y = 3\). To find the constant of variation, you substitute these values into the relationship formula \(y = kx\). This formula serves as a crucial part of solving direct variation problems, as it helps uncover \(k\), showing how \(x\) and \(y\) interact.
Remember, in direct variation, if you double \(x\), \(y\) doubles as well, provided that the relationship doesn’t change. This linearity is maintained by the value of \(k\).

Direct Variation Equation

The direct variation equation is at the heart of problems involving a linear relationship between two variables. The equation \(y = kx\) represents how each value of \(y\) corresponds directly with \(x\) scaled by the constant \(k\).
In our example, after calculating the constant of variation, we obtained \(k = 4\). Thus, the direct variation equation becomes \(y = 4x\). This means that for each unit increase in \(x\), \(y\) increases by 4 units.
The equation provides a simplified model of a direct relationship and can be used for predictions. For instance, if \(x = 1\), \(y\) would be 4 based on this equation. Recognizing and writing direct variation equations allows you to effortlessly model and understand linear relationships.

Solving for Constants

Solving for the constant of variation \(k\) is an essential step in tasks involving direct variation. You need to manipulate the initial formula \(y = kx\) by substituting the known values of \(x\) and \(y\).
In solving the exercise, the given values were \(x = \frac{3}{4}\) and \(y = 3\). By substituting these into the equation, \(3 = k * \frac{3}{4}\), you introduce an equation with one variable \(k\). To isolate \(k\), you simply divide both sides by \(\frac{3}{4}\), resulting in \(k = 4\). This simple operation unlocks the relationship between the two variables.
Successfully finding \(k\) empowers you to construct the equation \(y = 4x\) and make calculations accordingly. Solving for constants ensures that you properly adjust equations to reflect real-world data and relationships.