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:

$$y=k{x}^2$$

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:

$$k=y{x}^2$$

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 $k=y{x}^2$ 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.