To determine if the equation [tex]\( y = 5x - \square \)[/tex] represents a direct variation, we need to analyze the equation and compare it to the standard form of a direct variation equation.
A direct variation equation has the form:
[tex]\[ y = kx \][/tex]
where [tex]\( k \)[/tex] is a constant.
1. Original Equation:
[tex]\[ y = 5x - \square \][/tex]
2. Condition for Direct Variation:
For the equation to represent a direct variation, the term containing [tex]\(\square\)[/tex] must not alter the direct proportionality between [tex]\( y \)[/tex] and [tex]\( x \)[/tex]. This means the constant term must be zero to match the form [tex]\( y = kx \)[/tex].
3. Substitute 0 in the Box:
If Lydia puts 0 in the box, the equation becomes:
[tex]\[ y = 5x - 0 \][/tex]
which simplifies to:
[tex]\[ y = 5x \][/tex]
4. Analysis of the Modified Equation:
The modified equation [tex]\( y = 5x \)[/tex] is indeed of the form [tex]\( y = kx \)[/tex], where [tex]\( k = 5 \)[/tex]. Therefore, this represents a direct variation.
By comparing the modified equation to the standard form of a direct variation equation, we can conclude:
- If she puts 0 in the box, she would have a direct variation.
Thus, among the provided explanations, "If she puts 0 in the box she would have a direct variation" is the correct explanation.
