Question

$Z$ varies jointly with $x$ and $y$ when $x = 7$ and $y = 2$ , $z = 28$ . How do we write the function that models each variation and then find $z$ when $x = 6$ and $y = 4$ ?

Answer 1

Hint: To solve this question, first we should find the function of given values of $x$ , $y$ and $z$ by knowing the relation between one variable with the other two. And, then with the help of function, we can find the value of $z$ when the value of $x$ and $y$ changes.

Complete step by step solution:
As we know that, in the direction of the z-axis, $k$ is a unit vector.
When, $x = 7$ , $y = 2$
and, $z = 28 = 2 \times 7 \times 2 = 2.x.y$
So, the function here is $z = 2.x.y$
We know that the function has the form
$z = kxy$ , so
$\therefore k = \dfrac{z}{{xy}}$
If $x = 7$ , $y = 2$ and $z = 28$ ;
$ \Rightarrow k = \dfrac{{28}}{{7 \times 2}}$
$ \Rightarrow k = \dfrac{{28}}{{14}}$
$\therefore k = 2$
Now, we have the function:
$z = 2.x.y$
If $x = 6$ and $y = 4$ ; then to find the value of $z$ ; we have to put the value of $x$ and $y$ in the function.
Hence,
$z = 2.x.y\, = 2 \times 6 \times 4 = 48$

Note:- When you start studying algebra, you will also study how two (or more) variables can relate to each other specifically. The cases you’ll study are: Direct Variation, where one variable is a constant multiple of another Inverse or Indirect Variation, where when one of the variables builds, the other one reduces (their item is constant).