The constant of proportionality

Direct and inverse proportions are fundamental concepts in algebra. In these problems, you can find the constant of proportionality \( k \) when you are given values for \( x \) and \( y \). This is crucial for understanding the relationship between the two variables and solving for unknown quantities in future problems.

1. Direct Proportion:

In direct proportion, the relationship between \( x \) and \( y \) is given by the equation:

\[ y = kx \]

Where \( k \) is the constant of proportionality. To find \( k \) when \( x \) and \( y \) are given, rearrange the equation:

\[ k = \frac{y}{x} \]

2. Finding \( k \) Given \( x \) and \( y \) (Direct Proportion):

When you are given values for \( x \) and \( y \), substitute them into the formula \( k = \frac{y}{x} \) to find the constant of proportionality \( k \).

Example: If \( x = 4 \) and \( y = 12 \), find \( k \):

\[ k = \frac{12}{4} = 3 \]

So, the constant of proportionality \( k = 3 \).

3. Inverse Proportion:

In inverse proportion, the relationship between \( x \) and \( y \) is given by the equation:

\[ y = \frac{k}{x} \]

Where \( k \) is the constant of proportionality. To find \( k \) when \( x \) and \( y \) are given, rearrange the equation:

\[ k = xy \]

4. Finding \( k \) Given \( x \) and \( y \) (Inverse Proportion):

When you are given values for \( x \) and \( y \), substitute them into the formula \( k = xy \) to find the constant of proportionality \( k \).

Example: If \( x = 6 \) and \( y = 4 \), find \( k \):

\[ k = 6 \times 4 = 24 \]

So, the constant of proportionality \( k = 24 \).

5. Using \( k \) in Further Calculations:

Once you have found \( k \), you can use it to calculate unknown values of \( x \) or \( y \) in future problems. For example, in direct proportion, if you know \( k \) and a new value of \( x \), you can find \( y \). In inverse proportion, if you know \( k \) and a new value of \( x \), you can find \( y \).

6. Direct Proportion with Squared Terms:

In some problems, \( y \) may be directly proportional to the square of \( x \). The relationship is expressed as:

\[ y = kx^2 \]

To find \( k \), rearrange the equation:

\[ k = \frac{y}{x^2} \]

Example: If \( x = 3 \) and \( y = 18 \), find \( k \):

\[ k = \frac{18}{3^2} = \frac{18}{9} = 2 \]

So, the constant of proportionality \( k = 2 \).

7. Inverse Proportion with Squared Terms:

In some problems, \( y \) may be inversely proportional to the square of \( x \). The relationship is expressed as:

\[ y = \frac{k}{x^2} \]

To find \( k \), rearrange the equation:

\[ k = yx^2 \]

Example: If \( x = 2 \) and \( y = 6 \), find \( k \):

\[ k = 6 \times 2^2 = 6 \times 4 = 24 \]

So, the constant of proportionality \( k = 24 \).

8. Summary:

• In direct proportion, \( y = kx \), and to find \( k \), use \( k = \frac{y}{x} \).
• In inverse proportion, \( y = \frac{k}{x} \), and to find \( k \), use \( k = xy \).
• If the relationship involves squared terms, use \( k = \frac{y}{x^2} \) for direct proportion and \( k = yx^2 \) for inverse proportion to find \( k \).
• Once you have found \( k \), you can use it to solve for unknown values of \( x \) or \( y \) in future problems.

By understanding how to find \( k \) in both direct and inverse proportion, you can solve many real-world problems involving rates, distances, and quantities.