Question 199078
Let n=number of nickels and d=number of dimes


Since "Joe has a collection of nickels and dimes worth $6.05", this tells that {{{0.05n+0.10d=6.05}}}. Multiply EVERY term by 100 (to make every number whole) to get {{{5n+10d=605}}}


So the first equation is {{{5n+10d=605}}}


Also, "the number of dimes was doubled and the number of nickels was decreased by 10, the value would be $9.85" means that {{{0.05(n-10)+0.10(2d)=9.85}}}. Distribute to get {{{0.05n-0.5+0.2d=9.85}}}. Multiply EVERY term by 100 to make every number positive. So we now get {{{5n-50+20d=985}}}. Add 50 to both sides to get {{{5n+20d=1035}}}


So the second equation is {{{5n+20d=1035}}}





So we have the system of equations:


{{{system(5n+10d=605,5n+20d=1035)}}}



{{{-1(5n+10d)=-1(605)}}} Multiply the both sides of the first equation by -1.



{{{-5n-10d=-605}}} Distribute and multiply.



So we have the new system of equations:

{{{system(-5n-10d=-605,5n+20d=1035)}}}



Now add the equations together. You can do this by simply adding the two left sides and the two right sides separately like this:



{{{(-5n-10d)+(5n+20d)=(-605)+(1035)}}}



{{{(-5n+5n)+(-10d+20d)=-605+1035}}} Group like terms.



{{{0n+10d=430}}} Combine like terms.



{{{10d=430}}} Simplify.



{{{d=(430)/(10)}}} Divide both sides by {{{10}}} to isolate {{{d}}}.



{{{d=43}}} Reduce.



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{{{-5n-10d=-605}}} Now go back to the first equation.



{{{-5n-10(43)=-605}}} Plug in {{{d=43}}}.



{{{-5n-430=-605}}} Multiply.



{{{-5n=-605+430}}} Add {{{430}}} to both sides.



{{{-5n=-175}}} Combine like terms on the right side.



{{{n=(-175)/(-5)}}} Divide both sides by {{{-5}}} to isolate {{{n}}}.



{{{n=35}}} Reduce.



So our answers are {{{n=35}}} and {{{d=43}}}.



This means that there are 35 nickels and 43 dimes