Question 202047
I'll do the first two problems to get you going in the right direction.




# 1


Let 


x = number of bags of oranges purchased

y = number of bags of apples purchased



Because "15 bags of fruit were purchased", we know that {{{x+y=15}}}


Since the oranges are $5 a bag and the apples are $2 a bag, and a total cost was $33, this means that {{{5x+2y=33}}}




So we have the system of equations:



{{{system(x+y=15,5x+2y=33)}}}



{{{x+y=15}}} Start with the first equation.



{{{y=15-x}}} Subtract {{{x}}} from both sides.



{{{y=-x+15}}} Rearrange the terms.



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{{{5x+2(-x+15)=33}}} Now plug in {{{y=-x+15}}} into the second equation.



{{{5x-2x+30=33}}} Distribute.



{{{3x+30=33}}} Combine like terms on the left side.



{{{3x=33-30}}} Subtract {{{30}}} from both sides.



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



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



{{{x=1}}} Reduce.



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Since we know that {{{x=1}}}, we can use this to find {{{y}}}.



{{{x+y=15}}} Go back to the first equation.



{{{1+y=15}}} Plug in {{{x=1}}}.



{{{y=15-1}}} Subtract {{{1}}} from both sides.



{{{y=14}}} Combine like terms on the right side.




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Answer:




So the solutions are {{{x=1}}} and {{{y=14}}}.



This means that 1 bag of oranges and 14 bags of apples were purchased



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# 2



Let 



x = number of cars washed

y = number of vans washed



Because "They washed 45 vehicles", we know that {{{x+y=45}}}



Also, since "They washed cars for $5 each and vans for $7 each" and "made $243", we get that {{{5x+7y=243}}}



So we have the system of equations:



{{{system(x+y=45,5x+7y=243)}}}



{{{x+y=45}}} Start with the first equation.



{{{y=45-x}}} Subtract {{{x}}} from both sides.



{{{y=-x+45}}} Rearrange the terms.



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{{{5x+7(-x+45)=243}}} Now plug in {{{y=-x+45}}} into the second equation.



{{{5x-7x+315=243}}} Distribute.



{{{-2x+315=243}}} Combine like terms on the left side.



{{{-2x=243-315}}} Subtract {{{315}}} from both sides.



{{{-2x=-72}}} Combine like terms on the right side.



{{{x=(-72)/(-2)}}} Divide both sides by {{{-2}}} to isolate {{{x}}}.



{{{x=36}}} Reduce.



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Since we know that {{{x=36}}}, we can use this to find {{{y}}}.



{{{x+y=45}}} Go back to the first equation.



{{{36+y=45}}} Plug in {{{x=36}}}.



{{{y=45-36}}} Subtract {{{36}}} from both sides.



{{{y=9}}} Combine like terms on the right side.





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Answer:



So the solutions are {{{x=36}}} and {{{y=9}}}.



This means that they washed 36 cars and 9 vans