Sure! Let's determine which of the given functions have [tex]\( y \)[/tex] varying directly with [tex]\( x \)[/tex].
A function [tex]\( y \)[/tex] varies directly with [tex]\( x \)[/tex] if it can be expressed in the form:
[tex]\[ y = kx \][/tex]
where [tex]\( k \)[/tex] is a constant.
Let's evaluate each function one by one.
1. [tex]\( y = 4x + 1 \)[/tex]:
- This function is in the form [tex]\( y = kx + b \)[/tex], where [tex]\( b \neq 0 \)[/tex].
- Since there is a constant term [tex]\( +1 \)[/tex], it is not in the form [tex]\( y = kx \)[/tex].
- Therefore, [tex]\( y \)[/tex] does not vary directly with [tex]\( x \)[/tex] in this function.
2. [tex]\( y = 2x - 7 \)[/tex]:
- This function is in the form [tex]\( y = kx + b \)[/tex], where [tex]\( b \neq 0 \)[/tex].
- Since there is a constant term [tex]\( -7 \)[/tex], it is not in the form [tex]\( y = kx \)[/tex].
- Therefore, [tex]\( y \)[/tex] does not vary directly with [tex]\( x \)[/tex] in this function.
3. [tex]\( y = 0.5x \)[/tex]:
- This function is in the form [tex]\( y = kx \)[/tex], where [tex]\( k = 0.5 \)[/tex].
- There is no constant term added or subtracted.
- Therefore, [tex]\( y \)[/tex] varies directly with [tex]\( x \)[/tex] in this function.
4. [tex]\( y = 0x \)[/tex]:
- This function is in the form [tex]\( y = kx \)[/tex], where [tex]\( k = 0 \)[/tex].
- Even though [tex]\( k \)[/tex] is zero, the form [tex]\( y = kx \)[/tex] is maintained.
- Therefore, [tex]\( y \)[/tex] varies directly with [tex]\( x \)[/tex] in this function.
Based on this analysis, the functions where [tex]\( y \)[/tex] varies directly with [tex]\( x \)[/tex] are:
[tex]\[ y = 0.5x \][/tex]
[tex]\[ y = 0x \][/tex]
Thus, the indices of these functions in the given list are 3 and 4, respectively.
The final functions where [tex]\( y \)[/tex] varies directly with [tex]\( x \)[/tex] are found at the positions:
[tex]\[ (3, 4) \][/tex]
