Problem 18 - Algebra 1

Question

The variables x and y vary directly. Use the given values to write an equation that relates x and y. $$x=24, y=16$$

Step-by-Step Solution

Answer

The equation that relates x and y is \(y = 2/3x\).

Step 1: Identify Constant of Variation

It is found by dividing y by x. So, \(k = y/x = 16/24 = 2/3\). This means that the variables x and y have a constant of variation that equals 2/3, implying that y varies directly as 2/3 of x.

Step 2: Write the Equation

Now, substituting the value of k into the direct variation equation \(y = kx\), we get \(y = 2/3x\). This is the equation that relates x and y.

Key Concepts

Constant of Variation

In direct variation, the "constant of variation" refers to the unchanging value that relates two variables in a proportional relationship. It is symbolized by "k". To determine this constant, we use the formula:\[ k = \frac{y}{x} \]This formula shows how much one variable changes as the other variable changes at a consistent rate. For example, if you know that when \(x = 24\), \(y = 16\), then:

• Divide the value of \(y (16)\) by \(x (24)\).
• This gives \(k = \frac{16}{24} = \frac{2}{3}\).

The result, \(\frac{2}{3}\), tells us that for every unit increase in \(x\), \(y\) increases by \(\frac{2}{3}\) units. By knowing this constant, we can easily predict how changes in one variable might impact the other.

Linear Relationship

A "linear relationship" occurs when two variables are related through a constant rate or ratio, making a straight line when graphed. In the context of direct variation, this means that the relationship between the variables \(x\) and \(y\) can be expressed as a linear equation:\[ y = kx \]In our example, the constant of variation \(k\) is \(\frac{2}{3}\), so the linear relationship is represented as:\[ y = \frac{2}{3}x \]This equation tells us a few key things:

• It shows a direct relationship where \(y\) increases as \(x\) increases, tracing a linear path.
• The slope of this "line" on a graph is \(\frac{2}{3}\), representing the rate of change.

In essence, this linear equation helps us to easily calculate the value of \(y\) for any given value of \(x\) and vice versa.

Proportionality

"Proportionality" in a mathematical sense means that two variables increase or decrease at the same rate. In a direct variation, this can be summarized by stating that \(y\) is proportional to \(x\). The equation \(y = kx\) underscores this idea.Here are some important aspects of proportionality:

• In a direct variation, the ratio \(\frac{y}{x}\) is always equal to the constant of variation \(k\).
• If \(x\) doubles, then \(y\) also doubles, maintaining the ratio \(\frac{2}{3}\) in this case.
• Graphically, a proportional relationship is represented by a straight line through the origin, showing that \(y\) is proportional to \(x\) with no additive constant.

Understanding proportional relationships allows us to solve problems related to scaling, resizing, and predicting outcomes based on known measures. This occurs frequently in real-world applications like physics, economics, and everyday scenarios where ratios are significant.