Question 884284
RT=J Basic Equation, rate, time, job.


Let m = rate for 1 man
Let w = rate for 1 woman

Each work session is build on a term RT but specifically these are in parts of 1/3, 2/9, and the remaining part of the job to make one whole job.


{{{(4m*4+10w*4)=1/3}}}
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{{{6m*2+12w*2=2/9}}}
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How many more men, working 3 more day?
Let q = the additional number of men
{{{(6+q)m*3+12w*3=1-(1/3+2/9)}}}
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BE SURE YOU UNDERSTAND EACH OF THOSE THREE EQUATIONS BEFORE CONTINUING.  The first is accounting for the amount of work in the first four days; the second is accounting for the work in the next two day; the last is accounting for the last portion of the work using unknown q increased men in that last three days of work.  


Observe how those THREE equations use THREE unknown variables.  This appears to NOT be a linear system.  This should not be a major difficulty because the first two equations form a system of TWO equations in TWO unknowns, m and w, to be found first.



SIMPLIFY THE WHOLE SYSTEM:
{{{16m+40w=1/3}}}
{{{highlight_green(48m+120w=1)}}}
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{{{12m+24w=2/9}}}
{{{6m+12w=1/9}}}
{{{highlight_green(54m+108w=1)}}}
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{{{(6+q)m*3+12w*3=1-(1/3+2/9)}}}
{{{18m+3mq+36w=9/9-(3/9+2/9)}}}
{{{18m+3mq+36w=(4)/9}}}
and because we really will later want to solve for q,
{{{3mq=4/9-18m-36w}}}
{{{q=(4/9-18m-36w)/(3m)}}}
{{{highlight_green(q=4/27-6m-12w)}}}
The simplified equations are shown outlined in {{{highlight_green(GREEN)}}}.


SOLVE THE SYSTEM:
First solve for m and w in the first two equations as a separate subsystem.  I suggest substitution method because the coefficients are not very convenient for using elimination method.
USE the values for m and w found, and compute the value for q.


The rest of that "SOLVE THE SYSTEM" work is undone here but you should (need ) to do it.