Question 162958
Let x=# of $32 books and y=# of $44 books


Since "a local library bought 54 books", this means that {{{x+y=54}}}. Also, since "some cost $32 each and some cost $44 each. the total cost of the books was $1968", this tells us that {{{32x+44y=1968}}}






So we have the system of equations:


{{{system(x+y=54,32x+44y=1968)}}}



Let's solve the system by use of substitution




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





So let's isolate y in the first equation


{{{x+y=54}}} Start with the first equation



{{{y=54-x}}}  Subtract {{{x}}} from both sides



{{{y=-x+54}}} Rearrange the equation


---------------------


Since {{{y=-x+54}}}, we can now replace each {{{y}}} in the second equation with {{{-x+54}}} to solve for {{{x}}}




{{{32x+44highlight((-x+54))=1968}}} Plug in {{{y=-x+54}}} into the second equation. In other words, replace each {{{y}}} with {{{-x+54}}}. Notice we've eliminated the {{{y}}} variables. So we now have a simple equation with one unknown.




{{{32x+(44)(-x)+(44)(54)=1968}}} Distribute {{{44}}} to {{{-x+54}}}



{{{32x-44x+2376=1968}}} Multiply



{{{-12x+2376=1968}}} Combine like terms on the left side



{{{-12x=1968-2376}}}Subtract 2376 from both sides



{{{-12x=-408}}} Combine like terms on the right side



{{{x=(-408)/(-12)}}} Divide both sides by -12 to isolate x




{{{x=34}}} Divide






-----------------First Answer------------------------------



So the first part of our answer is: {{{x=34}}}










Since we know that {{{x=34}}} we can plug it into the equation {{{y=-x+54}}} (remember we previously solved for {{{y}}} in the first equation).




{{{y=-x+54}}} Start with the equation where {{{y}}} was previously isolated.



{{{y=-(34)+54}}} Plug in {{{x=34}}}



{{{y=-34+54}}} Multiply



{{{y=20}}} Combine like terms 




-----------------Second Answer------------------------------



So the second part of our answer is: {{{y=20}}}










-----------------Summary------------------------------


So our answers are:



{{{x=34}}} and {{{y=20}}}



This means that 34 $32 books and 20 $44 books were purchased