If Y Y Y Varies Directly As X X X , And Y Y Y Is 48 When X X X Is 6, Which Expression Can Be Used To Find The Value Of Y Y Y When X X X Is 2?A. Y = 48 6 ( 2 Y=\frac{48}{6}(2 Y = 6 48 ( 2 ]B. Y = 6 48 ( 2 Y=\frac{6}{48}(2 Y = 48 6 ( 2 ]C.
Understanding Direct Variation
Direct variation is a type of relationship between two variables, where one variable is a constant multiple of the other. 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 mathematically as:
y = kx
where y is the dependent variable, x is the independent variable, and k is the constant of variation.
Given Information
We are given that y varies directly as x, and y is 48 when x is 6. This means that we can use the direct variation equation to find the value of k.
y = kx
48 = k(6)
To find the value of k, we can divide both sides of the equation by 6:
k = 48/6
k = 8
Finding the Value of y When x is 2
Now that we have found the value of k, we can use the direct variation equation to find the value of y when x is 2.
y = kx
y = 8(2)
y = 16
Expression to Find the Value of y
We are asked to find the expression that can be used to find the value of y when x is 2. Based on the direct variation equation, we can see that the expression is:
y = kx
Substituting the value of k, we get:
y = 8x
When x is 2, the expression becomes:
y = 8(2)
y = 16
Conclusion
In conclusion, the expression that can be used to find the value of y when x is 2 is:
y = 8x
This expression represents the direct variation relationship between y and x, where y is a constant multiple of x.
Answer
The correct answer is:
C. y = 8x
Discussion
Direct variation is a fundamental concept in mathematics, and it has numerous applications in various fields, such as physics, engineering, and economics. Understanding direct variation and its applications can help us solve problems and make predictions in a wide range of situations.
Example Problems
- If y varies directly as x, and y is 24 when x is 3, find the value of y when x is 5.
- If y varies directly as x, and y is 36 when x is 4, find the value of y when x is 6.
- If y varies directly as x, and y is 48 when x is 6, find the value of y when x is 2.
Solutions
- y = kx
24 = k(3)
k = 24/3
k = 8
y = 8(5)
y = 40
- y = kx
36 = k(4)
k = 36/4
k = 9
y = 9(6)
y = 54
- y = kx
48 = k(6)
k = 48/6
k = 8
y = 8(2)
y = 16
Final Thoughts
Frequently Asked Questions
Q: What is direct variation?
A: Direct variation is a type of relationship between two variables, where one variable is a constant multiple of the other. In other words, as one variable increases or decreases, the other variable also increases or decreases at a constant rate.
Q: How is direct variation represented mathematically?
A: Direct variation can be represented mathematically as:
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 (k)?
A: The constant of variation (k) is a number that represents the constant rate at which the dependent variable (y) changes in response to changes in the independent variable (x).
Q: How do I find the value of k?
A: To find the value of k, you can use the direct variation equation and substitute the values of y and x. For example:
y = kx
48 = k(6)
k = 48/6
k = 8
Q: What is the relationship between y and x in direct variation?
A: In direct variation, y is a constant multiple of x. This means that as x increases or decreases, y also increases or decreases at a constant rate.
Q: Can you give an example of direct variation in real life?
A: Yes, direct variation can be seen in many real-life situations, such as:
- The cost of a product increases directly with the quantity purchased.
- The speed of a car increases directly with the amount of fuel used.
- The temperature of a room increases directly with the amount of heat added.
Q: How do I use direct variation to solve problems?
A: To use direct variation to solve problems, you can follow these steps:
- Write the direct variation equation: y = kx
- Substitute the values of y and x: y = k(6)
- Solve for k: k = 48/6
- Substitute the value of k back into the equation: y = 8x
- Use the equation to solve for y when x is a specific value.
Q: What are some common mistakes to avoid when working with direct variation?
A: Some common mistakes to avoid when working with direct variation include:
- Not using the correct units for the variables.
- Not substituting the correct values into the equation.
- Not solving for k correctly.
- Not using the correct equation to solve for y.
Q: Can you give some examples of direct variation problems?
A: Yes, here are some examples of direct variation problems:
- If y varies directly as x, and y is 24 when x is 3, find the value of y when x is 5.
- If y varies directly as x, and y is 36 when x is 4, find the value of y when x is 6.
- If y varies directly as x, and y is 48 when x is 6, find the value of y when x is 2.
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 relationship is proportional, meaning that as one variable increases or decreases, the other variable also increases or decreases at a constant rate.
- The problem can be represented mathematically using the direct variation equation: y = kx.
Q: What are some real-world applications of direct variation?
A: Direct variation has many real-world applications, including:
- Physics: The motion of objects, the force of gravity, and the energy of particles.
- Engineering: The design of bridges, buildings, and other structures.
- Economics: The cost of goods and services, the supply and demand of products, and the inflation rate.
- Biology: The growth and development of living organisms, the spread of diseases, and the behavior of populations.