Problem 32 - Algebra 1

Question

The variables x and y vary inversely. Use the given values to write an equation that relates x and y. $$x=10.5, y=7$$

Step-by-Step Solution

Answer

The equation that relates x and y is \(xy = 73.5\).

Step 1: Understand Inverse Variation

In mathematics, inverse variation (or inverse proportionality) means that when one variable increases, the other decreases, and vice versa. This fact is usually expressed in the equation \(xy = k\), where k is the constant of variation.

Step 2: Determine the Constant of Variation

Knowing the concept of inverse variation from the previous step, start by plugging in the given values of x and y into the equation \(xy = k\). Here, \(x = 10.5\) and \(y = 7\) so it becomes: \(10.5 * 7 = k\). Calculate this and find that \(k = 73.5\).

Step 3: Formulate the Equation

Now that the constant of variation (k) is known, the inverse variation equation can be written as \(xy = 73.5\).

Key Concepts

Constant of Variation

When we talk about the constant of variation in the context of inverse variation, we are referring to a fixed numerical value that remains unchanged. This constant represents the product of the two variables involved. In an inverse variation, the product of the variables \(x\) and \(y\) is always equal to \(k\), the constant of variation. For instance, if you know the values of \(x\) and \(y\), you can compute \(k\) by multiplying them together.

In the exercise given, where \(x = 10.5\) and \(y = 7\), calculating \(k\) helps us understand their inverse relationship. By doing the math, we multiply 10.5 by 7 to get 73.5, which is the constant of variation \(k\). This constant will not change no matter how \(x\) and \(y\) individually fluctuate.

Equation Formulation

Creating an equation that captures the relationship between variables \(x\) and \(y\) in an inverse variation can be simple once you know the constant \(k\). With inverse variation, the formula \(xy = k\) is key.

In the exercise, after determining that the constant \(k\) is 73.5, the next step is straightforward. We put together the equation by setting \(xy\) equal to the constant: \(xy = 73.5\). This equation helps to illustrate the rule that as \(x\) increases, \(y\) decreases to maintain the same product, and vice versa.

Variables Relationship

Understanding the relationship between the variables \(x\) and \(y\) is crucial in inverse variation. In this type of mathematical relationship, one variable increases while the other decreases. This is why it's called 'inverse.'

Here, since \(xy = k\), you can see that if \(x\) grows larger, \(y\) must shrink to keep the equation balanced. If \(x\) becomes smaller, \(y\) will need to increase. This dynamic mirrors a see-saw action, where the movement of one side affects the other directly.

• The constant \(k\) binds this relationship.
• Each time you change either \(x\) or \(y\), both values recalibrate so that their product remains \(k\).

In practical terms, this helps us to predict one variable's value based on the other's position.