Answer:
1/2
Step-by-step explanation:
x is directly proportional to y (this means y will go on top because of the directly part) and inversely proportional to z (this means z will go on bottom due to the inversely part).
There is a constant k such that:
[tex]x=k \cdot \frac{y}{z}[/tex].
Contant means it will never change. It will not care what (x,y,z) you use, it will remain the same.
We will use the first point to find k and then that k will still be there no matter what (x,y,z) they give you.
We have (1/2 , 3/4 , 2/3) is on our graph of the equation:
[tex]x=k \cdot \frac{y}{z}[/tex].
Insert the numbers:
[tex]\frac{1}{2}=k \cdot \frac{\frac{3}{4}}{\frac{2}{3}}[/tex]
Multiply both sides by [tex]\frac{2}{3}[/tex]:
[tex]\frac{1}{2}\cdot \frac{2}{3}=k \cdot \frac{3}{4}[/tex]
Simplify left hand side:
[tex]\frac{1}{3}=k \cdot \frac{3}{4}[/tex]
Multiply both sides by 4:
[tex]\frac{4}{3}=k \cdot 3[/tex]
Multiply both sides by 1/3 (or you can say divide by 3):
[tex]\frac{4}{9}=k[/tex]
So k=4/9 no matter the (x,y,z).
[tex]x=\frac{4}{9} \cdot \frac{y}{z}[/tex]
We are asked to find x given y=7/8 and z=7/9.
Input these numbers:
[tex]x=\frac{4}{9} \cdot \frac{\frac{7}{8}}{\frac{7}{9}}[/tex]
Change the division to multiplication:
[tex]x=\frac{4}{9} \cdot \frac{7}{8} \cdot \frac{9}{7}[/tex]
I see a 7 on top and bottom that I can cancel:
[tex]x=\frac{4}{9} \cdot \frac{1}{8} \cdot \frac{9}{1}[/tex]
I see a 9 on top and bottom that I can cancel:
[tex]x=\frac{4}{1} \cdot \frac{1}{8} \cdot \frac{1}{1}[/tex]
Let's go ahead and multiply and reduce more later if we can.
Multiply straight across on top.
Multiply straight across on bottom.
[tex]x=\frac{4}{8}[/tex]
Divide top and bottom by 4:
[tex]x=\frac{1}{2}[/tex]
