Problem 13 - Elementary Algebra

Question

The relationship of \(x\) and \(y\) is an inverse variation. When \(x=2, y=10\). a. Find the constant of proportionality, \(k\). b. Write an equation that represents this inverse variation. c. Find \(y\) when \(x=5\).

Step-by-Step Solution

Answer

a. k = 20. b. \[ y = \frac{20}{x} \]. c. y = 4.

Step 1: Identify the relationship

Inverse variation means the product of two variables is a constant. In this case, the relationship is given by the equation \[ x \times y = k \].

Step 2: Find the constant of proportionality

To find the constant of proportionality (\( k \)), substitute the given values \( x = 2 \) and \( y = 10 \) into the inverse variation equation:\[ k = x \times y = 2 \times 10 = 20 \].

Step 3: Write the inverse variation equation

Use the constant of proportionality (\( k = 20 \)) to write the equation that represents the inverse variation:\[ x \times y = 20 \]. Or equivalently,\[ y = \frac{20}{x} \].

Step 4: Find \( y \) when \( x = 5 \)

Substitute \( x = 5 \) into the inverse variation equation to find \( y \):\[ y = \frac{20}{5} = 4 \].

Key Concepts

constant of proportionality

The constant of proportionality, often represented by the symbol \(k\), is a key element in understanding inverse variation. In simple terms, it is the value that remains constant when two variables are inversely related. For any two variables \(x\) and \(y\), if their product always equals a constant, then we say they are inversely proportional.
In the given exercise, we are provided with values for \(x\) and \(y\):
When \(x = 2\), \(y = 10\).
To find \(k\), simply multiply these values together:
\[ k = x \times y = 2 \times 10 = 20 \]
This means the constant of proportionality is 20.

inverse variation equation

An inverse variation equation shows the relationship between two variables where one variable increases as the other decreases. In mathematical terms, the product of the variables always equals the constant of proportionality \(k\).
The general form is:
\[ x \times y = k \]
For our exercise, after finding the constant of proportionality as 20, we can write the inverse variation equation as:
\[ x \times y = 20 \]
This can also be rearranged to solve for \(y\) as a function of \(x\):
\[ y = \frac{20}{x} \]
This form is helpful when you need to calculate one variable given the other.

solving for variables

Solving for variables in an inverse variation involves finding one variable when you know the other. Using the inverse variation equation \( x \times y = k \) or its rearranged form \( y = \frac{20}{x} \), you can easily find unknown values.
In the exercise's final step, we need to find \(y\) when \(x = 5\).
Substitute \(x = 5\) into the equation:
\[ y = \frac{20}{5} = 4 \]
This shows that \(y\) equals 4 when \(x\) is 5.
By following these steps, you can solve inverse variation problems by identifying the constant of proportionality, writing the appropriate equation, and substituting known values to find unknown variables.