Question 1012279
{{{x}}}= number of true/false questions answered
{{{y}}}= number of multiple choice questions answered
Since "A correct answer to a true false question earns 10 points" and
"A correct answer to a multiple choice question earns 12 points",
{{{score=10x+12y}}} , which is a linear function on two variables.
We have some constraints that limit the domain of that function.
{{{x>=0}}} and {{{y>=0}}} are implicit constraints.


"Students have ... to answer to answer at most 18 questions of their choice",
{{{x+y<=18}}} , 
is an explicit constraint that sounds strange to a rat-race accustomed American,
who would expect that instead students would be required to rush to answer as many questions as possible,
and maybe to answer a minimum number of questions, as in {{{x+y>=18}}} .
Maybe the students are only given credit for {{{18}}} questions in order to encourage them to take time and work carefully, instead of rushing to answer as many questions as possible.
The expected pace, "it takes, on average, 3 minutes to answer a true/false question and 4 minutes to answer a multiple choice question", means that
{{{3x+4y}}}= time (in minutes) needed to answer {{{x}}} true/false questions and {{{y}}} multiple choice questions.
That leads to another explicit constraint of the problem, because the student can use at most {{{60}}} minutes (1 hour).
That constraint can be written as
{{{3x+4y<=60}}} .
The line {{{3x+4y=60}}} obviously passes through {{{system(x=0,y=18)}}} and {{{system(y=0,x=20)}}} , so we graph it as {{{drawing(300,250,-3,27,-3,24,grid(1),
line(-4,18,24,-3)
)}}} , and since {{{system(x=0,y=0)}}} is a solution to {{{3x+4y<=60}}} ,
the graph for {{{3x+4y<=60}}} , {{{drawing(300,250,-3,27,-3,24,grid(1),
graph(300,250,-3,27,-3,24,3x+4y<=60)
)}}} , includes the origin, the point (0,0) . 
Working in the same manner, we find that the constraint {{{x+y<=18}}} can be graphed as {{{drawing(300,250,-3,27,-3,24,grid(1),
graph(300,250,-3,27,-3,24,x+y<=18)
)}}} ,
while {{{x+y>=18}}} would be graphed as {{{drawing(300,250,-3,27,-3,24,grid(1),
graph(300,250,-3,27,-3,24,x+y>=18)
)}}} .
 
With the constraints {{{system(x>=0,y>=0,3x+4y<=60,x+y<=18)}}} , we have four inequalities that determine a feasibility space that is a convex polygon. It is
{{{drawing(300,250,-3,27,-3,24,grid(1),
green(line(0,15,12,6)),green(line(12,6,18,0)),
green(line(0,0,0,15)),green(line(0,0,18,0))
)}}} , with vertices at (0,0) , (0,15) , (12,6) , and (18,0) .
 
An extreme value (maximum or minimum) of linear function, such as {{{score=10x+12y}}} , defined on a domain that is a {{{score=10x+12y}}} ,
occurs at a vertex of the domain, or all along an edge between two vertices,
so we only need to check the scores for the vertices.
At (0,0) , obviously {{{score=0}}} .
At (0,15) , {{{system(x=0,y=15)}}} , and {{{score=10*0+12*15=180}}} .
At (18,0) , {{{system(x=18,y=0)}}} , and {{{score=10*18+12*0=180}}} .
At (12,6) , {{{highlight(system(x=12,y=6))}}} , {{{score=10*12+12*6=120+72=192}}} , and the student gets the greatest possible score.