x is directly proportional to y and inversely proportional to z. if x= 1/2 when y = 3/4 and z=2/3, find x when y= 7/8 and z = 7/9

Accepted Solution

Answer:1/2Step-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]