Question

ydia writes the equation below with a missing value.

y = 5 x minus box

She puts a value in the box and says that the equation represents a direct variation. Which explains whether the equation could represent a direct variation?
If she puts 0 in the box she would have a direct variation.
If she puts 5 in the box she would have a direct variation.
The equation is not of the form y = k x, so it cannot represent a direct variation.
The equation has a minus sign, so it cannot represent a direct variation.

Answer 1

To determine whether the equation \( y = 5x - \text{box} \) represents a direct variation, we need to recall what direct variation means. An equation represents a direct variation if it can be put in the form \( y = kx \), where \( k \) is a constant. In other words, \( y \) is directly proportional to \( x \).

1. If \( \text{box} = 0 \):
- The equation becomes \( y = 5x - 0 \), which simplifies to \( y = 5x \). This is of the form \( y = kx \) (with \( k = 5 \)), so this does represent a direct variation.

2. If \( \text{box} = 5 \):
- The equation becomes \( y = 5x - 5 \). This is not of the form \( y = kx \) because it has a constant term that does not equal 0. Therefore, this does not represent a direct variation.

3. The statement "The equation is not of the form \( y = k x \), so it cannot represent a direct variation" is true, but it applies only when the box has a value other than 0.

4. The statement "The equation has a minus sign, so it cannot represent a direct variation" is not necessarily correct. A minus sign does not exclude the possibility of direct variation; it depends on whether the equation can be simplified to the form \( y = kx \).

Based on this analysis, the correct explanation is:

**If she puts 0 in the box, she would have a direct variation.**