Problem 11 - Algebra 1: Concepts and Skills

Question

Suppose \(y=6\) when \(x=4 .\) For the given type of variation, find an equation that relates \(x\) and \(y .\) $$\text{\(x\) and \(y\) vary inversely.}$$

Step-by-Step Solution

Answer

The equation that relates \(x\) and \(y\) is \(xy = 24\).

Step 1: Identify the variation type

The problem states that \(x\) and \(y\) vary inversely. This implies that as \(x\) increases, \(y\) decreases and vice versa. This type of relationship can be represented by the general mathematical equation \(xy = k\), where \(k\) is a constant. Our task is to find the value of this constant.

Step 2: Plug in known values

We know from the problem that when \(x = 4\), \(y = 6\). We substitute these values into the general equation for inverse variation. So, \(4*6 = k\).

Step 3: Calculate the constant

By performing the multiplication, we find that \(k = 24\).

Step 4: Write the final equation for the inverse variation

Now that we have the constant \(k\), we can write the specific equation of inverse variation that relates \(x\) and \(y\). The final equation is \(xy = 24\). Note that the final equation conforms to the general form \(xy = k\), as expected for problems involving inverse variation.

Key Concepts

Inverse Relationship

An inverse relationship is a type of connection between two variables where as one increases, the other decreases. This is a distinctive pattern that contrasts with direct relationships, where variables move in the same direction.
In the context of our problem, if we observe that \(x\) and \(y\) vary inversely, this means that when the value of \(x\) goes up, the value of \(y\) must go down in order to maintain a constant product. Conversely, when \(x\) decreases, \(y\) must increase. Imagine a seesaw: when one end goes up, the other must come down.

• Inverse relationships are often seen in physics, economics, and other areas involving proportional change.
• They help us understand balance within systems, like ensuring a fixed total amount, which remains constant as individual parts adjust.

Algebraic Equation

An algebraic equation is a mathematical statement that uses letters (variables) to represent numbers in relationships. These variables are arranged in equations to show how they interact.
In terms of inverse variation, this specific relationship is typically expressed as \(xy = k\), where \(x\) and \(y\) are the variables, and \(k\) is a constant that does not change as long as the variables maintain their inversely proportional relationship.

• In our example, by plugging the values \(x = 4\) and \(y = 6\) into the equation, we can find \(k\).
• Performing the calculation gives \(xy = 4 \times 6 = 24\).

This is how algebraic equations can be applied to decipher the connection between two categorically different quantities, reflecting their inverse nature.

Constant of Variation

The constant of variation is a specific number that remains unchanged in the equation of an inverse relationship. It signifies the consistent product of the variables \(x\) and \(y\) in our equation \(xy = k\).
To find this constant, simply use known values of \(x\) and \(y\), substitute them into the inverse variation equation, and solve for \(k\). In the example provided, when \(x = 4\) and \(y = 6\), substituting into the equation \(4 \times 6 = k\) yields \(k = 24\).

• This constant acts as the 'anchor' maintaining the inverse relationship.
• No matter what values \(x\) and \(y\) take (as long as they conform to the inverse relationship), their product will always equal this constant, in this case, 24.

Understanding and calculating the constant of variation is essential, as it is key to defining and predicting the behavior of the variables involved, particularly through changes in their values.