Question 850678
Use all rates as Jobs per Days.  Notice that C worker is given enough direct information for a rate value.  Start with the rate for C, and create rate equations for B & C, and for A & B.


Rate for C:  {{{1/50}}} jobs per day.


Let x = time for A to do 1 job.
Let y = time for B to do 1 job.


First sentence tells, rate for A is equal to the combined rate of B and C.
{{{1/x=1/y+1/50}}}
LCD is 50xy.
{{{50*xy/x=50*xy/y+50*xy/50}}}
{{{50y=50x+xy}}}


Second sentence tells, A and B together do the job in 10 days:
{{{(1/x+1/y)=1/10}}}
LCD is 10xy.
{{{10xy/x+10xy/y=10xy/10}}}
{{{10y+10x=xy}}}


The system to solve for x and y is:
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{{{50y=50x+xy}}}
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{{{10y+10x=xy}}}
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Solving each of those for xy, and then equating each expression's formula for xy gives {{{50y-50x=10y+10x}}}
simplifying to 
{{{2y=3x}}}
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Solving the linear equation for a variable, trying y, {{{y=3x/3}}};
Picking either system equation, trying {{{10y+10x=xy}}}, and substitute the expression for y found,
{{{highlight_green(10(3x/2)+10x=x(3x/2))}}}
Multiply left and right members by 2, and be sure to obtain this equation:
{{{3x^2-50=0}}}  -----do not try to divide either sides by x thinking that to be simplification; it's a common beginners mistake.
{{{highlight_green(x(3x-50)=0)}}}
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The meaningful answer for x is that {{{highlight(x=50/3=16&2/3)}}}
.. and use either system equation to find or solve for y.  You can also use the linear {{{highlight(y=(3/2)x)}}} found earlier.