there are no values of z that make the equation true.
there were 12 questions worth 3 points
let x = 2 point questions
let y = 3 point questions
what we know:
x + y = 20 (questions)
so to solve for x we tweak our equation:
x = 20 - y
now we need an equation to explain out question set up:
2x (2 point questions)
3y (2 point questions)
2x + 3y = 52 (52 being our total possible points)
now we take our x = 20 - y and substitute it into our 2x + 3y = 52 equation.
2(20 - y) + 3y = 52
next we distribute the 2 through the parenthesis:
40 - 2y + 3y = 52
now, gather the like terms:
40 + y = 52
subtract 40 from both sides:
y = 12
revisit x + y = 20 and sub 12 in for y:
x + 12 = 20
x = 8
8 2-point questions would be 16 points
12 3-point questions would be 36 points
16 + 36 = 52 (which matches the max value of our test)
so you want to prove . notice that for , we have
if , we have
and we know , so will always be a multiple of 14 and we're done.
for 12 increasing...