Problem 53 - Algebra 1

Question

The variables x and y vary directly. Use the given values to write an equation that relates x and y. $$x=-13, y=-52$$

Step-by-Step Solution

Answer

The equation that relates x and y is \(y = 4x\).

Step 1: Find the constant of variation

In direct variation, the constant of variation (k) can be found by dividing y by x. So, let's divide -52 by -13. \(k = -52 / -13 = 4\)

Step 2: Write the equation

Now that it was found that k equals 4, the equation relating x and y in direct variation is \(y = 4x\).

Key Concepts

Constant of Variation

In mathematics, when two variables are directly proportional, they have a special relationship that can be described with a constant known as the "constant of variation." This constant is a key part of direct variation, also known as direct proportionality. To find the constant of variation, one must determine the ratio of the two variables that are directly proportional.
In the given example, the variables are \(x = -13\) and \(y = -52\). The constant of variation, \(k\), is calculated as follows:

• Divide \(y\) by \(x\) to find \(k\).
• \(k = \frac{-52}{-13} = 4\)

Here, \(k = 4\) tells us that for every unit increase in \(x\), \(y\) increases by four units. This constant remains the same, as long as the relationship between \(x\) and \(y\) is direct.
Understanding this concept is essential as it forms the basis of constructing the equation that links \(x\) and \(y\).

Equation Formulation

Once the constant of variation is discovered, forming the equation that describes the relationship between the variables in a direct variation is straightforward. This equation is essential because it precisely portrays how the variables interact.
The general form of a direct variation equation is \(y = kx\), where \(k\) is the constant of variation.
In our example:

• We previously found that \(k = 4\).
• Therefore, the direct variation equation is \(y = 4x\).

This equation shows that \(y\) changes proportionally with \(x\), scaling by a factor of 4. It is helpful for predicting \(y\) values when any \(x\) value is given, as long as the direct variation relationship holds true.

Variables in Direct Proportionality

Direct proportionality involves understanding the relationship between the variables \(x\) and \(y\). When \(x\) and \(y\) vary directly, their relationship can be expressed mathematically, and it implies that as one variable increases or decreases, the other does so at a constant rate.
Some key points about variables in direct proportionality include:

• The ratio \(\frac{y}{x}\) remains constant, which is the constant \(k\).
• If \(x\) doubles, \(y\) will double, and if \(x\) is halved, \(y\) is halved, maintaining a consistent relationship.
• In our specific problem, \(y\) is four times each value of \(x\), due to \(k = 4\).

This makes it easy to predict how changes in one variable affect the other, using the equation \(y = kx\). Thus, having a firm grasp of variables in direct proportionality aids greatly in comprehending how mathematical models project real-world phenomena.