We would start by assigning the variable s to be the number of soft serve cones bought and h to the number of hard pack cones bought.
So, the word problem has to be broken up into 2 separate equations to solve the problem.
One equation will be how many total cones were sold?
This answer can be found in the word problem as 79.
Therefore, s + h = 79
The second equation will deal with the amount of money spent.
The soft serve cones were $2.00 each, so the amount of money from soft serve would be s(2.00) or 2s.
The hard pack cones were $0.75 each, so the amount of money from hard pack would be h(0.75) or 0.75h
The equation, knowing from the word problem, that the total money would be $106.75 is as follows:
2s + 0.75h = 106.75
In order to solve this, you could use graphing, or a graphing calculator to see where these two equations would intersect, giving you the values for s and h that would add to 79 and give you a total of $106.75.
You could also solve this by Elimination of one of the variables, where you could manipulate the coefficient of one of the variables (probably s, would be easiest) to be equal and opposite values and eliminate this variable so you could solve for the other variable (ie. 1 equation, 1 unknown value)
A third way to solve this would be by substitution. Solve, say the first equation for s or h and then substitute that expression into the other equation and solve that for the unknown variable.
By substitution, normally the most difficult for the math, but easiest for students to see would give us the first equation, solved for s as
s = (79 - h)
this makes the second equation: 2s + 0.75h = 106.75
2(79 - h) +0.75h = 106.75 (Substitution)
158 - 2h + 0.75h = 106.75 (Distribution)
158 + (-1.25)h = 106.75 (Combine Terms)
(-1.25)h = 106.75 - 158 = (-51.25) (Isolate the variable h)
h = (-51.25)/(-1.25) = 41
So 41 hard packed cones served and 79-41 or 38 soft serve cones served.
Using graphing technology, like Desmos.com can be a great tool for a student to see the answer as well, very quickly.
