x = number of people who received compensation.
y = the amount of compensation each received.
t = the total compensation received by x number of people.
your main equation becomes:
x * y = t
if there were 6 people more, then each would have received 12,500 less.
the number of people becomes x+6 and the compensation each received becomes y-12,500. the total remains the same.
your equation for that becomes:
(x+6) * (y-12500) = t
if there were 5 people less, then each would have received 15,000 more.
the number of people becomes x-5 and the compensation each received becomes y+15,000. the total remains the same.
your equation for that becomes:
(x-5) * (y+15000) = t
you have 3 equations that need to be solved simultaneously.
this means the same value of x, y, and t solve each equation.
those equations are:
x * y = t which can be shown as xy = 5
(x+6) * (y-12500) = t
(x-5) * (y+15000) = t
simplify each of these equations by performing the multiplication indicated to get:
xy = t
xy - 12500x + 6y - 75000 = t
xy + 15000x - 5y - 75000 = t
since xy = t, you can replace xy by t in the second and third equations to get:
xy = t
t - 12500x + 6y - 75000 = t
t + 15000x - 5y - 75000 = t
leave the first equation as is and just work with the second and third equation for now.
those equations are:
t - 12500x + 6y - 75000 = t
t + 15000x - 5y - 75000 = t
subtract t from both sides of each of these equations to get:
-12500x + 6y - 75000 = 0
15000x - 5y - 75000 = 0
you now have two equations in two unknowns.
you want to eliminate the y so you can solve for x.
alternatively, you want to eliminate the x so you can solve for y.
we'll eliminate the y so we can solve for x.
multiply both sides of the first equation by 5 and multiply both sides of the second equation by 6 to get:
-62500x + 30y - 375000 = 0
90000x - 30y - 450000 = 0
add the equations together to get:
27500x - 825000 = 0
add 825000 to both sides of this equation to get:
27500x = 825000
divide both sides of this equation by 27500 to get:
x = 825000 / 27500
solve for x to get:
x = 30
go back to one of the original equations that started this sequence and replace x with 30 and solve for y.
one of the original equations is:
-12500x + 6y - 75000 = 0
replace x with 30 to get:
-12500 * 30 + 6y - 75000 = 0
simplify to get:
-375000 + 6y - 75000 = 0
combine like terms to get:
-450000 + 6y = 0
add 450000 to both sides of this equation to get:
6y = 450000
divide both sides of this equation by 6 to get:
y = 75000.
since x = 30 and y = 75000, and xy = t, looks like 30 families would receive 75,000 each for a total compensation of 2,250,000.
so the total compensation is equal to 2,250,000 dollars
if there were 6 more families, then each would have received 12,500 dollars less.
36 * (75,000 - 12,500) = 36 * 62,500 = 2,250,000.
this checks out ok.
total compensation remains the same.
if there were 5 less families, then each would have received 15,000 more.
25 * (75,000 + 15,000) = 25 * 90,000 = 2,250,000
this checks out ok.
total compensation remains the same.
based on these calculations:
the number of families that got relief is 30.
the total compensation is 2,250,000 dollars.
you started with three equations in three unknowns.
you used substitution to reduce the problem to two equations in two unknowns.
you used elimination to reduce the problem to one equation in one unknown.
you were then able to solve for that unknown and then use the value of that unknown to solve for the other unknowns.
to solve this graphically using the three equations would have required graphing in 3 dimensions.
once you were able to eliminate one of the unknowns and one of the equations, you could have solved this graphically using the two equations in two unknowns graphical method.
the equations that could have been graphed are:
-12500x + 6y - 75000 = 0
15000x - 5y - 75000 = 0
the graph would have looked like this:
the intersection of the two lines is the solution.
both equations intersect when x = 30 and y = 75000.