If { Y $}$ Varies Directly As { X $}$, And { Y $}$ Is 18 When { X $}$ Is 5, Which Expression Can Be Used To Find The Value Of { Y $}$ When { X $}$ Is 11?A. [$ Y = \frac{5}{18}(11)
Direct variation is a fundamental concept in mathematics that describes the relationship between two variables. In this scenario, we are given that y } varies directly as <span class="katex-error" title="ParseError', got 'EOF' at end of input: " style="color$, which means that as x } increases, <span class="katex-error" title="ParseError', got 'EOF' at end of input: " style="color$ also increases at a constant rate.
The Formula for Direct Variation
The formula for direct variation is given by:
{ y = kx $}$
where { k $}$ is the constant of variation, and { x $}$ and { y $}$ are the variables.
Given Information
We are given that { y $}$ is 18 when { x $}$ is 5. Using this information, we can find the value of the constant of variation { k $}$.
Finding the Constant of Variation
To find the value of { k $}$, we can substitute the given values into the formula:
{ 18 = k(5) $}$
Solving for { k $}$, we get:
{ k = \frac{18}{5} $}$
The Expression for Finding { y $}$
Now that we have found the value of { k $}$, we can use it to find the expression for { y $}$ when { x $}$ is 11.
{ y = \frac{18}{5}(11) $}$
This expression can be used to find the value of { y $}$ when { x $}$ is 11.
Simplifying the Expression
To simplify the expression, we can multiply the numerator and denominator by 11:
{ y = \frac{18}{5} \times \frac{11}{1} $}$
{ y = \frac{198}{5} $}$
Conclusion
In conclusion, the expression that can be used to find the value of { y $}$ when { x $}$ is 11 is { y = \frac{198}{5} $}$. This expression is derived from the formula for direct variation and the given information that { y $}$ is 18 when { x $}$ is 5.
Why is Direct Variation Important?
Direct variation is an important concept in mathematics because it helps us understand the relationship between two variables. It has numerous applications in real-life situations, such as:
- Physics: Direct variation is used to describe the relationship between distance, speed, and time.
- Economics: Direct variation is used to describe the relationship between supply and demand.
- Biology: Direct variation is used to describe the relationship between population growth and environmental factors.
Real-World Examples of Direct Variation
Here are some real-world examples of direct variation:
- Speed and Distance: The distance traveled by a car is directly proportional to its speed. If the speed of the car increases, the distance traveled also increases.
- Supply and Demand: The price of a product is directly proportional to its demand. If the demand for a product increases, the price also increases.
- Population Growth: The population of a city is directly proportional to its growth rate. If the growth rate of a city increases, the population also increases.
Common Misconceptions about Direct Variation
Here are some common misconceptions about direct variation:
- Direct Variation is the Same as Proportionality: Direct variation is not the same as proportionality. Proportionality is a specific type of direct variation where the constant of variation is equal to 1.
- Direct Variation is Only Used in Simple Situations: Direct variation is not only used in simple situations. It can be used to describe complex relationships between variables.
- Direct Variation is Only Used in Mathematics: Direct variation is not only used in mathematics. It has numerous applications in real-life situations.
Conclusion
In conclusion, direct variation is an important concept in mathematics that describes the relationship between two variables. It has numerous applications in real-life situations and is used to describe complex relationships between variables. By understanding direct variation, we can better understand the world around us and make more informed decisions.
Final Answer
Q: What is direct variation?
A: Direct variation is a mathematical relationship between two variables where one variable is a constant multiple of the other variable. It is a type of linear relationship where the ratio of the two variables remains constant.
Q: What is the formula for direct variation?
A: The formula for direct variation is:
{ y = kx $}$
where { k $}$ is the constant of variation, and { x $}$ and { y $}$ are the variables.
Q: How do I find the constant of variation?
A: To find the constant of variation, you can use the given information to set up an equation. For example, if { y $}$ is 18 when { x $}$ is 5, you can set up the equation:
{ 18 = k(5) $}$
Solving for { k $}$, you get:
{ k = \frac{18}{5} $}$
Q: What is the difference between direct variation and proportionality?
A: Direct variation and proportionality are related but distinct concepts. Direct variation is a general term that describes a linear relationship between two variables, while proportionality is a specific type of direct variation where the constant of variation is equal to 1.
Q: Can direct variation be used to describe complex relationships between variables?
A: Yes, direct variation can be used to describe complex relationships between variables. While it is often used to describe simple relationships, it can also be used to describe more complex relationships where the constant of variation is not equal to 1.
Q: What are some real-world examples of direct variation?
A: Some real-world examples of direct variation include:
- Speed and Distance: The distance traveled by a car is directly proportional to its speed.
- Supply and Demand: The price of a product is directly proportional to its demand.
- Population Growth: The population of a city is directly proportional to its growth rate.
Q: What are some common misconceptions about direct variation?
A: Some common misconceptions about direct variation include:
- Direct Variation is the Same as Proportionality: Direct variation is not the same as proportionality. Proportionality is a specific type of direct variation where the constant of variation is equal to 1.
- Direct Variation is Only Used in Simple Situations: Direct variation is not only used in simple situations. It can be used to describe complex relationships between variables.
- Direct Variation is Only Used in Mathematics: Direct variation is not only used in mathematics. It has numerous applications in real-life situations.
Q: How do I use direct variation to solve problems?
A: To use direct variation to solve problems, you can follow these steps:
- Identify the variables: Identify the variables involved in the problem and determine which one is the independent variable and which one is the dependent variable.
- Set up the equation: Set up the equation using the formula for direct variation.
- Find the constant of variation: Find the constant of variation using the given information.
- Solve for the unknown variable: Solve for the unknown variable using the equation and the constant of variation.
Q: What are some tips for working with direct variation?
A: Some tips for working with direct variation include:
- Make sure to identify the variables correctly: Make sure to identify the variables correctly and determine which one is the independent variable and which one is the dependent variable.
- Use the correct formula: Use the correct formula for direct variation and make sure to set up the equation correctly.
- Check your work: Check your work to make sure that you have solved the problem correctly.
Conclusion
In conclusion, direct variation is a mathematical relationship between two variables where one variable is a constant multiple of the other variable. It has numerous applications in real-life situations and is used to describe complex relationships between variables. By understanding direct variation, you can better understand the world around you and make more informed decisions.