Hi, Jane. The "how" in the last sentence gives the clue for what we are looking for. It indicates the missing information and thus the variables, as such:
Let x = the number of 3-point shots and y = the number of 2-point shots.
Equation 1 - points: 3x + 2y = 30
Equation 2 - shots: x + y = 11
Addition Method
(1) Arbitrarily choose a variable to cancel, we will select y. Multiple equation 2 by -2. The new system is
3x + 2y = 30
-2x - 2y = -22
When added vertically becomes
-> 1x +0y = 8
-> x = 8
(2) Arbitrarily substitute x = 8 into either equation then solve for y. We will choose the first, which becomes,
3(8) + 2y = 30
-> 24 + 2y = 30
-> 2y = 30 - 24
-> 2y = 6
-> y = 3
Substitution Method
(1) Arbitrarily isolate a "naked" variable, will chose the x in equation 2.
x + y = 11
-> x = 11 - y
(2) Substitute into the other equation, and solve for the remaining variable.
3(11 - y) + 2y = 30
-> 3*11 -3y + 2y = 30
-> 33 -3y + 2y = 30
-> 33 -y = 30
-> 33 - 30 = y
-> 3 = y
(3) Arbitrarily substitute y = 3 into any equation.
-> x = 11 - y
-> x = 11 - 3
-> x = 8
∴ Using either method our solution of (x, y) = (8, 3) tells us Alan scored eight 3-point shots and three 2-point shots.
Additionally, using two different methods that give the same result counts as doing a check step. To one manually, substitute the final solution into both original equations.
-> 3(8) + 2(3) ?= 30
-> 24 + 6 ?= 30
-> 30 = 30
&
-> 8 + 3 ?= 11
-> 11 = 11
