Problem 12 - Elementary Algebra

Question

The relationship of \(x\) and \(y\) is an inverse variation. When \(x=3, y=6\). a. Find the constant of proportionality, \(k\). b. Write an equation that represents this inverse variation. c. Find \(y\) when \(x=9\).

Step-by-Step Solution

Answer

The constant of proportionality is 18. The equation is \( y = \frac{18}{x} \). When x=9, y=2.

Step 1: Determine the relationship

Since this is an inverse variation, the relationship between x and y can be defined as: \[ y = \frac{k}{x} \] where k is the constant of proportionality.

Step 2: Find the constant of proportionality, k

Use the given values of x and y. When x = 3 and y = 6, substitute these values into the inverse variation equation to find k: \[ 6 = \frac{k}{3} \] Solve for k: \[ k = 6 \times 3 = 18 \]

Step 3: Write the equation

Substitute the value of k back into the inverse variation equation: \[ y = \frac{18}{x} \]

Step 4: Find y when x = 9

Substitute x = 9 into the equation to find y: \[ y = \frac{18}{9} \] Solve for y: \[ y = 2 \]

Key Concepts

constant of proportionality

Understanding the constant of proportionality is crucial to mastering inverse variation problems. This constant, often denoted as 𝑘, is a fixed number that links the two variables in an inverse variation relationship.
When two variables, say 𝑥 and 𝑦, vary inversely, their product is always the same. Mathematically, we express this as: \[ y = \frac{k}{x} \]
Here, 𝑘 is the constant of proportionality. To find 𝑘, you multiply the given values of 𝑥 and 𝑦. In our exercise, we are given that 𝑥=3 and 𝑦=6. Substituting these into the equation, we get: \[ 6 = \frac{k}{3} \]
Multiplying both sides of the equation by 3 to isolate 𝑘, we find: \[ k = 6 \times 3 = 18 \]
Thus, the constant of proportionality, 𝑘, is 18.

inverse variation equation

After determining the constant of proportionality, we can write the inverse variation equation that represents the relationship between 𝑥 and 𝑦. An inverse variation equation takes the form: \[ y = \frac{k}{x} \]
Given that 𝑘=18 from our calculation above, we substitute this value back into the equation, resulting in: \[ y = \frac{18}{x} \]
This equation tells us that for any value of 𝑥, we can find the corresponding value of 𝑦 by dividing 18 by 𝑥. This relationship is key in problems involving inverse variation.

solving for variables

Now that we have our inverse variation equation, solving for a variable is straightforward. Let's practice this with part (c) of our exercise: find 𝑦 when 𝑥=9. Using our inverse variation equation: \[ y = \frac{18}{9} \]
We simply substitute 9 for 𝑥 and solve the equation: \[ y = \frac{18}{9} = 2 \]
Therefore, when 𝑥=9, 𝑦=2.
To generalize, whenever you have an inverse variation equation and one of the variables, substitute the known value into the equation to find the unknown variable.
This method can be applied to any inverse variation problem, making it a versatile tool in algebra.