The Value Of $y$ Varies Directly With $x$, And \$y = 12$[/tex\] When $x = 4$.What Is $y$ When \$x = -8$[/tex\]?$y = [?\]
Understanding Direct Variation
Direct variation is a fundamental concept in mathematics where the value of one variable is directly proportional to the value of another variable. In other words, as one variable increases or decreases, the other variable also increases or decreases at a constant rate. This relationship can be represented by the equation y = kx, where y is the dependent variable, x is the independent variable, and k is the constant of variation.
The Given Problem
In this problem, we are given that the value of y varies directly with x, and y = 12 when x = 4. We need to find the value of y when x = -8.
Finding the Constant of Variation
To find the value of y when x = -8, we first need to find the constant of variation, k. We can do this by substituting the given values of x and y into the equation y = kx.
y = kx 12 = k(4)
Solving for k
To solve for k, we can divide both sides of the equation by 4.
k = 12/4 k = 3
The Equation of Direct Variation
Now that we have found the constant of variation, k, we can write the equation of direct variation as:
y = 3x
Finding the Value of y
Now that we have the equation of direct variation, we can find the value of y when x = -8 by substituting x = -8 into the equation.
y = 3(-8) y = -24
Conclusion
In this problem, we used the concept of direct variation to find the value of y when x = -8. We first found the constant of variation, k, by substituting the given values of x and y into the equation y = kx. Then, we used the equation of direct variation to find the value of y when x = -8.
Real-World Applications of Direct Variation
Direct variation has many real-world applications, including:
- Physics: The distance traveled by an object is directly proportional to the time it travels.
- Economics: The cost of a product is directly proportional to the quantity produced.
- Biology: The growth rate of a population is directly proportional to the size of the population.
Examples of Direct Variation
Here are a few examples of direct variation:
- y = 2x: This equation represents a direct variation where y is directly proportional to x with a constant of variation of 2.
- y = 5x: This equation represents a direct variation where y is directly proportional to x with a constant of variation of 5.
- y = 10x: This equation represents a direct variation where y is directly proportional to x with a constant of variation of 10.
Graphing Direct Variation
The graph of a direct variation is a straight line that passes through the origin. The slope of the line represents the constant of variation.
Conclusion
In this article, we discussed the concept of direct variation and how it can be used to solve problems in mathematics. We also explored the real-world applications of direct variation and provided examples of direct variation. Finally, we graphed direct variation and discussed the importance of understanding this concept in mathematics.
Final Answer
The final answer is:
Frequently Asked Questions About Direct Variation
Direct variation is a fundamental concept in mathematics that can be used to solve a wide range of problems. However, it can be a bit confusing at first, especially for those who are new to the concept. In this article, we will answer some of the most frequently asked questions about direct variation.
Q: What is direct variation?
A: Direct variation is a relationship between two variables where one variable is directly proportional to the other variable. This means that as one variable increases or decreases, the other variable also increases or decreases at a constant rate.
Q: How do I know if a problem is a direct variation problem?
A: To determine if a problem is a direct variation problem, look for the following characteristics:
- The problem involves a relationship between two variables.
- The problem states that one variable is directly proportional to the other variable.
- The problem can be represented by the equation y = kx, where y is the dependent variable, x is the independent variable, and k is the constant of variation.
Q: What is the constant of variation?
A: The constant of variation, k, is a number that represents the rate at which one variable changes in relation to the other variable. It is a key component of the equation y = kx and is used to determine the slope of the line that represents the direct variation.
Q: How do I find the constant of variation?
A: To find the constant of variation, k, you can use the following steps:
- Write the equation of direct variation in the form y = kx.
- Substitute the given values of x and y into the equation.
- Solve for k by dividing both sides of the equation by x.
Q: What is the equation of direct variation?
A: The equation of direct variation is y = kx, where y is the dependent variable, x is the independent variable, and k is the constant of variation.
Q: How do I graph a direct variation?
A: To graph a direct variation, follow these steps:
- Write the equation of direct variation in the form y = kx.
- Plot the point (0, 0) on the graph.
- Use the slope-intercept form of the equation to find the slope of the line.
- Plot additional points on the graph using the slope-intercept form of the equation.
- Draw a straight line through the points to represent the direct variation.
Q: What are some real-world applications of direct variation?
A: Direct variation has many real-world applications, including:
- Physics: The distance traveled by an object is directly proportional to the time it travels.
- Economics: The cost of a product is directly proportional to the quantity produced.
- Biology: The growth rate of a population is directly proportional to the size of the population.
Q: What are some examples of direct variation?
A: Here are a few examples of direct variation:
- y = 2x: This equation represents a direct variation where y is directly proportional to x with a constant of variation of 2.
- y = 5x: This equation represents a direct variation where y is directly proportional to x with a constant of variation of 5.
- y = 10x: This equation represents a direct variation where y is directly proportional to x with a constant of variation of 10.
Q: What are some common mistakes to avoid when working with direct variation?
A: Here are a few common mistakes to avoid when working with direct variation:
- Not checking the units: Make sure to check the units of the variables to ensure that they are consistent.
- Not using the correct equation: Make sure to use the correct equation of direct variation, y = kx.
- Not solving for k correctly: Make sure to solve for k correctly by dividing both sides of the equation by x.
Q: How can I practice working with direct variation?
A: Here are a few ways to practice working with direct variation:
- Work through examples: Work through examples of direct variation to practice solving for k and graphing the equation.
- Use online resources: Use online resources, such as Khan Academy or Mathway, to practice working with direct variation.
- Take practice quizzes: Take practice quizzes to test your understanding of direct variation.
Conclusion
In this article, we answered some of the most frequently asked questions about direct variation. We covered topics such as the definition of direct variation, how to find the constant of variation, and how to graph a direct variation. We also discussed some real-world applications of direct variation and provided examples of direct variation. Finally, we covered some common mistakes to avoid when working with direct variation and provided some tips for practicing working with direct variation.