Certainly! Let's address the problem step-by-step.
### Given:
- [tex]\( y \)[/tex] varies directly with [tex]\( x \)[/tex].
- When [tex]\( x = 6 \)[/tex], [tex]\( y = 42 \)[/tex].
#### a) Find the equation that relates [tex]\( y \)[/tex] and [tex]\( x \)[/tex].
Since [tex]\( y \)[/tex] varies directly with [tex]\( x \)[/tex], we can express this relationship as:
[tex]\[ y = kx \][/tex]
where [tex]\( k \)[/tex] is the constant of variation.
To determine the value of [tex]\( k \)[/tex], we use the given values [tex]\( y = 42 \)[/tex] and [tex]\( x = 6 \)[/tex]:
[tex]\[ 42 = k \cdot 6 \][/tex]
Solving for [tex]\( k \)[/tex]:
[tex]\[ k = \frac{42}{6} = 7 \][/tex]
Hence, the equation relating [tex]\( y \)[/tex] and [tex]\( x \)[/tex] is:
[tex]\[ y = 7x \][/tex]
#### b) What is [tex]\( y \)[/tex] when [tex]\( z = 33 \)[/tex]?
It seems like there's a slight error in notation because typically, [tex]\( y \)[/tex] and [tex]\( x \)[/tex] are used consistently. Assuming [tex]\( z \)[/tex] should actually be [tex]\( x \)[/tex], we are to find [tex]\( y \)[/tex] when [tex]\( x = 33 \)[/tex].
Using the equation [tex]\( y = 7x \)[/tex]:
[tex]\[ y = 7 \cdot 33 \][/tex]
Calculating the value:
[tex]\[ y = 231 \][/tex]
Therefore, when [tex]\( x = 33 \)[/tex], [tex]\( y = 231 \)[/tex].
In summary:
a) The equation relating [tex]\( y \)[/tex] and [tex]\( x \)[/tex] is [tex]\( y = 7x \)[/tex].
b) When [tex]\( x = 33 \)[/tex], [tex]\( y \)[/tex] is [tex]\( 231 \)[/tex].
