```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)}}}

{{{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.

=============================================================

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

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

<hr>

# 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)}}}

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

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

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

{{{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.

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

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.

=============================================================

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

This means that they washed 36 cars and 9 vans

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