Question 1119317
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If you don't have access to a tool that solves systems of equations for you, you are stuck with having to do a pencil-and-paper solution.<br>
For a general system of three linear equations, this can be very tedious.<br>
However, in a timed competitive exam (for example), there will usually be a way of combining the given equations to make the solution relatively easy.  So look at the coefficients in the given equations and see if you can see a way to combine them to perhaps eliminate one of the variables.<br>
The system of equations for this problem is<br>
{{{x+y+z = 290}}}
{{{11x+7y+2z = 1750}}}
{{{15x+8y+3z = 2280}}}<br>
In this system, I immediately see the "z", "2z" and "3z" in the three equations; if I add the first two equations and compare the resulting equation to the third, variable z will be eliminated.<br>
{{{12x+8y+3z = 2040}}}
{{{15x+8y+3z = 2280}}}
{{{3x = 240}}}
{{{x = 80}}}<br>
I got a bonus this time -- I eliminated TWO variables at once, allowing me to immediately find the value of the third.<br>
Plugging the value of x into the first two of the original equations, we get<br>
{{{y+z = 210}}}
{{{7y+2z = 870}}}<br>
Double the first equation and compare to the second to eliminate z:<br>
{{{2y+2z = 420}}}
{{{5y = 450}}}
{{{y = 90}}}
{{{90+z = 210}}}
{{{z = 120}}}<br>
Answer: x = 80 (adults); y = 90 (high school students); z = 120 (younger students)