Understanding Direct Variation

Oct 13, 2025 by pynkocean.com 33 views

Understanding Direct Variation

Direct variation is a relationship between two or more variables where one variable is a constant multiple of the other variable(s). In this case, we are dealing with a direct variation between three variables: ${$ y $}$, ${$ x $}$, and ${$ z $}$. This means that ${$ y $}$ varies directly as the product of ${$ x $}$ and ${$ z $}$.

Mathematical Representation

The direct variation relationship can be represented mathematically as:

${$ y = kxz $}$

where ${$ k $}$ is the constant of variation.

Given Values

We are given that ${$ y = 4 $}$ when ${$ x = 6 $}$ and ${$ z = 1 $}$. We can use this information to find the constant of variation ${$ k $}$.

Finding the Constant of Variation

To find the constant of variation, we can substitute the given values into the direct variation equation:

${$ 4 = k(6)(1) $}$

Simplifying the equation, we get:

${$ 4 = 6k $}$

Dividing both sides by 6, we get:

${$ k = \frac{4}{6} = \frac{2}{3} $}$

Direct Variation Equation with Constant

Now that we have found the constant of variation, we can write the direct variation equation with the constant:

${$ y = \frac{2}{3}xz $}$

Finding ${$ y $}$ when ${$ x = 7 $}$ and ${$ z = 4 $}$

We are asked to find ${$ y $}$ when ${$ x = 7 $}$ and ${$ z = 4 $}$. We can substitute these values into the direct variation equation with the constant:

${$ y = \frac{2}{3}(7)(4) $}$

Simplifying the equation, we get:

${$ y = \frac{2}{3}(28) $}$

Multiplying, we get:

${$ y = \frac{56}{3} $}$

Final Answer

Therefore, when ${$ x = 7 $}$ and ${$ z = 4 $}$, ${$ y = \frac{56}{3} $}$.

Conclusion

In this article, we have discussed direct variation with two variables and how to find the constant of variation. We have also used the direct variation equation with the constant to find ${$ y $}$ when ${$ x = 7 $}$ and ${$ z = 4 $}$. The direct variation equation with the constant is a powerful tool for solving problems involving direct variation.

Direct Variation with Two Variables: Key Takeaways

Direct variation is a relationship between two or more variables where one variable is a constant multiple of the other variable(s).
The direct variation relationship can be represented mathematically as ${$ y = kxz $}$, where ${$ k $}$ is the constant of variation.
To find the constant of variation, we can substitute the given values into the direct variation equation.
The direct variation equation with the constant can be used to find ${$ y $}$ when ${$ x $}$ and ${$ z $}$ are given.

Direct Variation with Two Variables: Practice Problems

If ${$ y $}$ varies directly as ${$ x $}$ and ${$ z $}$, and ${$ y = 9 $}$ when ${$ x = 3 $}$ and ${$ z = 2 $}$, find ${$ y $}$ when ${$ x = 4 $}$ and ${$ z = 3 $}$.
If ${$ y $}$ varies directly as ${$ x $}$ and ${$ z $}$, and ${$ y = 12 $}$ when ${$ x = 2 $}$ and ${$ z = 3 $}$, find ${$ y $}$ when ${$ x = 3 $}$ and ${$ z = 4 $}$.

Direct Variation with Two Variables: Solutions

To find ${$ y $}$ when ${$ x = 4 $}$ and ${$ z = 3 $}$, we can substitute these values into the direct variation equation with the constant:

${$ y = \frac{2}{3}(4)(3) $}$

Simplifying the equation, we get:

${$ y = \frac{2}{3}(12) $}$

Multiplying, we get:

${$ y = 8 $}$

To find ${$ y $}$ when ${$ x = 3 $}$ and ${$ z = 4 $}$, we can substitute these values into the direct variation equation with the constant:

${$ y = \frac{2}{3}(3)(4) $}$

Simplifying the equation, we get:

${$ y = \frac{2}{3}(12) $}$

Multiplying, we get:

${$ y = 8 $}$

Direct Variation with Two Variables: Conclusion

In this article, we have discussed direct variation with two variables and how to find the constant of variation. We have also used the direct variation equation with the constant to find ${$ y $}$ when ${$ x $}$ and ${$ z $}$ are given. The direct variation equation with the constant is a powerful tool for solving problems involving direct variation.