Question 148022
# 1

When "y varies directly as x", this means that as x increases y increases. Also as x decreases y decreases. So the first equation is {{{y=kx}}}. If let k be any number and you plug in values of x (say 1,2,3, etc), you'll notice that y will increase along with x.



Furthermore, when varies inversely as the square root of w, this means that as the square root of w increases y decreases. Also as square root of w decreases y increases. So the second equation is {{{y=k/sqrt(w)}}}.  


Now if you combine the two equations, you get 


{{{y=(kx)/(sqrt(w))}}}



So the entire sentence "y varies directly as x and inversely as the square root of w" translates to {{{y=(kx)/(sqrt(w))}}}

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# 2


{{{3^x=y}}} Start with the given equation.


{{{log(3,(y))=x}}} Rewrite the equation using the property: {{{b^y=x}}} ===> {{{log(b,(x))=y}}} 



So the equation in logarithmic form is {{{log(3,(y))=x}}}




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# 3



{{{log(10,(2^x))=y}}} Start with the given equation.



{{{10^y=2^x}}} Rewrite the equation using the property: {{{log(b,(x))=y}}} ====> {{{b^y=x}}}



So the equation in exponential form is {{{10^y=2^x}}} 



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# 4



{{{5x/6-x/3=1/6}}} Start with the given equation.



{{{6(5x/cross(6)-x/cross(3))=cross(6)(1/cross(6))}}} Multiply both sides by the LCD {{{6}}} to clear any fractions.



{{{5x-2x=1}}} Distribute and multiply.



{{{3x=1}}} Combine like terms on the left side.



{{{x=(1)/(3)}}} Divide both sides by {{{3}}} to isolate {{{x}}}.



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Answer:


So the answer is {{{x=1/3}}} 



Which approximates to {{{x=0.333}}}