Find The Constant Of Variation For The Relation And Use It To Write An Equation For The Statement. Then Solve The Equation.If $y$ Varies Directly As $x$ And $z$, And $y=40$ When $x=5$ And
Direct Variation with Two Variables: Finding the Constant of Variation and Writing an Equation
In this article, we will explore the concept of direct variation with two variables. Direct variation is a relationship between two or more variables where one variable is a constant multiple of the others. In this case, we have a relation where $y$ varies directly as $x$ and $z$. We will use this information to find the constant of variation and write an equation for the statement. Finally, we will solve the equation to find the value of $y$.
Direct variation is a type of linear relationship between two or more variables. It can be represented by the equation $y = kx$, where $k$ is the constant of variation. In this case, we have a relation where $y$ varies directly as $x$ and $z$. This means that $y$ is a constant multiple of $x$ and $z$.
To find the constant of variation, we can use the given information that $y=40$ when $x=5$ and $z=2$. We can substitute these values into the equation $y = kxz$ to solve for $k$.
So, the constant of variation is $k = 4$.
Now that we have found the constant of variation, we can write an equation for the statement. The equation is:
This equation represents the direct variation relationship between $y$, $x$, and $z$.
To solve the equation, we can substitute the given values of $x$ and $z$ into the equation and solve for $y$.
This confirms that the value of $y$ is indeed $40$ when $x=5$ and $z=2$.
In this article, we have explored the concept of direct variation with two variables. We have found the constant of variation and written an equation for the statement. Finally, we have solved the equation to find the value of $y$. This demonstrates the power of direct variation in modeling real-world relationships.
Direct variation with two variables has many practical applications in fields such as physics, engineering, and economics. For example, the force of gravity between two objects varies directly as the product of their masses and inversely as the square of the distance between them. This relationship can be represented by the equation $F = G\frac{m_1m_2}{r^2}$, where $F$ is the force of gravity, $G$ is the gravitational constant, $m_1$ and $m_2$ are the masses of the objects, and $r$ is the distance between them.
When working with direct variation with two variables, it is essential to remember the following tips and tricks:
- Always check the units of the variables to ensure that they are consistent.
- Use the given information to find the constant of variation.
- Write an equation for the statement using the constant of variation.
- Solve the equation to find the value of the variable.
By following these tips and tricks, you can master the concept of direct variation with two variables and apply it to real-world problems.
Q: What is direct variation with two variables? A: Direct variation with two variables is a relationship between two or more variables where one variable is a constant multiple of the others.
Q: How do I find the constant of variation? A: To find the constant of variation, use the given information to substitute the values into the equation and solve for the constant.
Q: What is the equation for direct variation with two variables? A: The equation for direct variation with two variables is $y = kxz$, where $k$ is the constant of variation.
Q: How do I solve the equation? A: To solve the equation, substitute the given values of $x$ and $z$ into the equation and solve for $y$.
By understanding direct variation with two variables, you can model real-world relationships and solve problems in various fields.