Question 123351

Let n= # of nickels, and d= # of dimes


Since "Jill has $3.50 in nickels and dimes", it can be represented as:



{{{0.05n+0.1d=3.50}}}



{{{100(0.05n+0.1d)=100(3.50)}}} Multiply both sides by 100 to move the decimal on each number 2 times. This will make every number a whole number.


{{{5n+10d=350}}} Distribute and multiply




So our first equation is: {{{5n+10d=350}}} 



Now since  Jill has 50 coins, this means that we have the second equation


{{{n+d=50}}}



So we have the system 



{{{system(5n+10d=350,n+d=50)}}}




Now in order to solve this system by using elimination/addition, we need to solve (or isolate) one variable. I'm going to solve for d.






In order to solve for one variable, we must eliminate the other variable. So if we wanted to solve for {{{d}}}, we would have to eliminate {{{n}}} (or vice versa).



So lets eliminate {{{n}}}. In order to do that, we need to have both {{{n}}} coefficients that are equal in magnitude but have opposite signs (for instance 2 and -2 are equal in magnitude but have opposite signs). This way they will add to zero. By adding to zero, they can be eliminated.




So to make the {{{n}}} coefficients equal in magnitude but opposite in sign, we need to multiply both {{{n}}} coefficients by some number to get them to an common number. So if we wanted to get {{{5}}} and {{{1}}} to some equal number, we could try to get them to the LCM.




Since the LCM of {{{5}}} and {{{1}}} is {{{5}}}, we need to multiply both sides of the top equation by {{{1}}} and multiply both sides of the bottom equation by {{{-5}}} like this:





{{{1(5n+10d)=1(350)}}}  Multiply the top equation (both sides) by {{{1}}}
{{{-5(n+d)=-5(50)}}}  Multiply the bottom equation (both sides) by {{{-5}}}





Distribute and multiply


{{{5n+10d=350}}}
{{{-5n-5d=-250}}}



Now add the equations together. In order to add 2 equations, group like terms and combine them


{{{(5n-5n)+(10d-5d)=350-250}}}


Combine like terms and simplify




{{{cross(5n-5n)+5d=100}}} Notice how the n terms cancel out





{{{5d=100}}} Simplify





{{{d=100/5}}} Divide both sides by {{{5}}} to isolate d





{{{d=20}}} Reduce




Now plug this answer into the top equation {{{5n+10d=350}}} to solve for x


{{{5n+10d=350}}} Start with the first equation




{{{5n+10(20)=350}}} Plug in {{{d=20}}}




{{{5n=350-200}}}Subtract 200 from both sides



{{{5n=150}}} Combine like terms on the right side



{{{n=(150)/(5)}}} Divide both sides by 5 to isolate n




{{{n=30}}} Divide





So our answer is

{{{n=30}}} and {{{d=20}}}




which means that Jill has 30 nickels and 20 dimes