b = number of bicycling injuries.
f = number of football injuries.
g + b + f = 3179
b = g + 800
f = b + 1267
these 3 equations need to be solved simultaneously.
replace b in the first equation and the third equation with g + 800 from the second equation to get:
g + g + 800 + f = 3179
b = g + 800
f = g + 800 + 1267
replace f in the first equation with g + 800 + 1267 from the third equation to get:
g + g + 800 + g + 800 + 1267 = 3179
b = g + 800
f = g + 800 + 1267
combine like terms to get:
3g + 2867 = 3179
b = g + 800
f = g + 2067
solve for g in the first equation to get g = (3179 - 2867) / 3 = 104
solve for b in the second equation to get b = 104 + 800 = 904
solve for f in the third equation to get f = 104 + 2067 = 2171
evaluate your original equations with these values to see if they hold true.
g + b + f = 3179 becomes 104 + 904 + 2171 = 3179 which is true.
b = g + 800 becomes 904 = 104 + 800 which is true.
f = b + 1267 becomes 2171 = 904 + 1267 which is true.
all original equations are true, therefore the solution can be assumed to be good.
you could also have solved this in the following manner, and gotten the same answer.
your original equations are:
g + b + f = 3179
b = g + 800
f = b + 1267
rearrange them so that the variables are on the left and the constants are on the right and fill in the missing variables with 0.
you would get:
g + b + f = 3179
-g + b + 0 = 800
0 - b + f = 1267
you would then eliminate b from the first two equations to get:
2g + f = 2379
you would then eliminate b from the second two equations to get:
-g + f = 2067
you would then eliminate f from these two equations to get:
3g = 312
you would then solve for g to get g = 104
from there, you can easily solve for b and f.
my worksheet for doing it this way is shown below:
in this worksheet:
E1, E2, and E3 are the original equations that have been re-arranged so that the variables are on the left side and the constant is on the right side of each equation.
if a variable is missing, then a 0 is placed where it would normally be.
i then subtracted E2 from E1 to get E4.
this eliminated the variable b from E4.
i then added E2 to E3 to get E5.
this also eliminated the variable b from E5.
i was left with two equations in two unknowns.
these were E4 and E5.
i then subtracted E5 from E4 to get E6.
this eliminated f and left me with one equation in one variable.
i then solve for g to get g = 104.
once i got that i was able to go back and solve for b and f.
the first method was by substitution.
the second method was by elimination.