Question 152717
a)


Let x=# of adults and y=# of students



Since "There were 1800 tickets sold", this means that {{{x+y=1800}}}. Also,  because the prices were "$8 for adults, $5 for students" which gave them "a total of $12 750", this means that {{{8x+5y=12750}}}


So we have the system:


{{{system(x+y=1800,8x+5y=12750)}}}



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b)




Start with the given system of equations:

{{{system(x+y=1800,8x+5y=12750)}}}



{{{-8(x+y)=-8(1800)}}} Multiply the both sides of the first equation by -8.



{{{-8x-8y=-14400}}} Distribute and multiply.



So we have the new system of equations:

{{{system(-8x-8y=-14400,8x+5y=12750)}}}



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



{{{(-8x-8y)+(8x+5y)=(-14400)+(12750)}}}



{{{(-8x+8x)+(-8y+5y)=-14400+12750}}} Group like terms.



{{{0x+-3y=-1650}}} Combine like terms. Notice how the x terms cancel out.



{{{-3y=-1650}}} Simplify.



{{{y=(-1650)/(-3)}}} Divide both sides by {{{-3}}} to isolate {{{y}}}.



{{{y=550}}} Reduce.



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{{{-8x-8y=-14400}}} Now go back to the first equation.



{{{-8x-8(550)=-14400}}} Plug in {{{y=550}}}.



{{{-8x-4400=-14400}}} Multiply.



{{{-8x=-14400+4400}}} Add {{{4400}}} to both sides.



{{{-8x=-10000}}} Combine like terms on the right side.



{{{x=(-10000)/(-8)}}} Divide both sides by {{{-8}}} to isolate {{{x}}}.



{{{x=1250}}} Reduce.



So our answer is {{{x=1250}}} and {{{y=550}}}.



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c)

Since  {{{x=1250}}} and {{{y=550}}}, this means that there were 1,250 adults and 550 students.



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d)


Check:


Let's check the first equation



{{{x+y=1800}}} Start with the first equation.



{{{1250+550=1800}}} Plug in {{{x=1250}}} and {{{y=550}}}.



{{{1800=1800}}} Add. Since this equation is true, this means that {{{x=1250}}} and {{{y=550}}} are solutions to the equation.



Now let's check the second equation



{{{8x+5y=12750}}} Start with the second equation.



{{{8*(1250)+5*(550)=12750}}} Plug in {{{x=1250}}} and {{{y=550}}}.



{{{10000+2750=12750}}} Multiply



{{{12750=12750}}} Add. Since this equation is true, this means that {{{x=1250}}} and {{{y=550}}} are solutions to the equation.



Since <font size="4"><b>both</b></font> equations of the system are true, this verifies our answer.