If $y$ Varies Directly As $x$, And $y$ Is 48 When $x$ Is 6, Which Expression Can Be Used To Find The Value Of $y$ When $x$ Is 2?A. $y = \frac{48}{6}(2)$ B. $y =

Mar 11, 2025 by ADMIN 177 views

**Direct Variation: Finding the Value of y**

Understanding Direct Variation

Direct variation is a type of relationship between two variables, x and y, where the value of y is directly proportional to the value of x. This means that as x increases, y also increases, and vice versa. The relationship between x and y can be represented by the equation y = kx, where k is the constant of proportionality.

Given Information

We are given that y varies directly as x, and y is 48 when x is 6. This information can be used to find the value of the constant of proportionality, k.

Finding the Constant of Proportionality

To find the value of k, we can use the given information and substitute it into the equation y = kx. We know that y is 48 when x is 6, so we can write:

48 = k(6)

To solve for 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 know the value of k, we can use it to find the value of y when x is 2. We can substitute x = 2 and k = 8 into the equation y = kx:

y = 8(2) y = 16

Evaluating the Answer Choices

Now that we have found the value of y when x is 2, we can evaluate the answer choices to see which one is correct.

A. y = 48/6(2) B. y = 8(2)

The correct answer is B. y = 8(2), which is equal to 16.

Conclusion

In this problem, we used the concept of direct variation to find the value of y when x is 2. We first found the value of the constant of proportionality, k, using the given information. Then, we used the value of k to find the value of y when x is 2. The correct answer is B. y = 8(2), which is equal to 16.

Direct Variation Formula

The direct variation formula is y = kx, where k is the constant of proportionality.

Finding the Constant of Proportionality

To find the value of k, we can use the given information and substitute it into the equation y = kx.

Example

Suppose y varies directly as x, and y is 48 when x is 6. To find the value of k, we can write:

48 = k(6)

To solve for 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 know the value of k, we can use it to find the value of y when x is 2. We can substitute x = 2 and k = 8 into the equation y = kx:

y = 8(2) y = 16

Evaluating the Answer Choices

Now that we have found the value of y when x is 2, we can evaluate the answer choices to see which one is correct.

A. y = 48/6(2) B. y = 8(2)

The correct answer is B. y = 8(2), which is equal to 16.

Conclusion

Direct Variation Problems

Direct variation problems can be solved using the direct variation formula, y = kx. To solve a direct variation problem, we need to find the value of k and then use it to find the value of y.

Example Problems

Suppose y varies directly as x, and y is 24 when x is 4. Find the value of y when x is 8.
Suppose y varies directly as x, and y is 36 when x is 6. Find the value of y when x is 9.

Solutions

To find the value of k, we can write:

24 = k(4)

To solve for k, we can divide both sides of the equation by 4:

k = 24/4 k = 6

Now that we know the value of k, we can use it to find the value of y when x is 8. We can substitute x = 8 and k = 6 into the equation y = kx:

y = 6(8) y = 48

To find the value of k, we can write:

36 = k(6)

To solve for k, we can divide both sides of the equation by 6:

k = 36/6 k = 6

Now that we know the value of k, we can use it to find the value of y when x is 9. We can substitute x = 9 and k = 6 into the equation y = kx:

Q: What is direct variation?

A: Direct variation is a type of relationship between two variables, x and y, where the value of y is directly proportional to the value of x. This means that as x increases, y also increases, and vice versa.

Q: How is direct variation represented mathematically?

A: Direct variation is represented mathematically by the equation y = kx, where k is the constant of proportionality.

Q: What is the constant of proportionality?

A: The constant of proportionality, k, is a value that represents the rate at which y changes in response to changes in x.

Q: How do I find the constant of proportionality?

A: To find the constant of proportionality, you can use the given information and substitute it into the equation y = kx. Then, solve for k.

Q: What if I have two points on the graph of a direct variation? How can I find the constant of proportionality?

A: If you have two points on the graph of a direct variation, you can use the formula k = (y2 - y1) / (x2 - x1) to find the constant of proportionality.

Q: How do I use the constant of proportionality to find the value of y when x is a given value?

A: To find the value of y when x is a given value, you can substitute the value of x and the constant of proportionality into the equation y = kx.

Q: What if I have a direct variation problem with a negative constant of proportionality? How do I solve it?

A: If you have a direct variation problem with a negative constant of proportionality, you can still use the equation y = kx to solve it. However, the value of y will be negative when x is positive, and vice versa.

Q: Can I have a direct variation problem with a zero constant of proportionality?

A: Yes, it is possible to have a direct variation problem with a zero constant of proportionality. In this case, the equation y = kx will be y = 0, and the value of y will always be zero.

Q: What are some real-world examples of direct variation?

A: Some real-world examples of direct variation include:

The relationship between the distance traveled and the time taken to travel a certain distance.
The relationship between the amount of money spent and the number of items purchased.
The relationship between the temperature and the amount of ice cream sold.

Q: How can I use direct variation to solve problems in real life?

A: You can use direct variation to solve problems in real life by identifying the variables involved and using the equation y = kx to find the value of y when x is a given value.

Direct Variation Practice Problems

Suppose y varies directly as x, and y is 24 when x is 4. Find the value of y when x is 8.
Suppose y varies directly as x, and y is 36 when x is 6. Find the value of y when x is 9.
Suppose y varies directly as x, and y is 48 when x is 6. Find the value of y when x is 2.
Suppose y varies directly as x, and y is 72 when x is 9. Find the value of y when x is 12.

Solutions

To find the value of k, we can write:

24 = k(4)

To solve for k, we can divide both sides of the equation by 4:

k = 24/4 k = 6

Now that we know the value of k, we can use it to find the value of y when x is 8. We can substitute x = 8 and k = 6 into the equation y = kx:

y = 6(8) y = 48

To find the value of k, we can write:

36 = k(6)

To solve for k, we can divide both sides of the equation by 6:

k = 36/6 k = 6

Now that we know the value of k, we can use it to find the value of y when x is 9. We can substitute x = 9 and k = 6 into the equation y = kx:

y = 6(9) y = 54

To find the value of k, we can write:

48 = k(6)

To solve for k, we can divide both sides of the equation by 6:

k = 48/6 k = 8

Now that we know the value of k, we can use it to find the value of y when x is 2. We can substitute x = 2 and k = 8 into the equation y = kx:

y = 8(2) y = 16

To find the value of k, we can write:

72 = k(9)

To solve for k, we can divide both sides of the equation by 9:

k = 72/9 k = 8

Now that we know the value of k, we can use it to find the value of y when x is 12. We can substitute x = 12 and k = 8 into the equation y = kx:

y = 8(12) y = 96