The first two steps in determining the solution set of the system of equations, y = -x2 + 4x + 12 and y=-3x + 24, algebraically
are shown in the table.
Step
Equation
Step 1
-x² + 4x + 12 =-3x+24
Step 2
x2 - 7x+12-0
Which represents the solution(s) of this system of equations?
(3, 15)
(-4, 36)
(3, 15) and (4, 12)
(-3, 33) and (-4,36)

(3, 15) and (4, 12)

Step-by-step explanation:

x2 - 7x+12 = 0 factors to (x - 4)(x - 3) = 0, and so x:  {3, 4}.

Since y= -3x + 24, y = 15 when x = 3 and y = 12 when x = 4.  The solutions are thus (3, 15) and (4, 12).  This is the third answer choice.

3. parallelogram

step-by-step explanation:

we turn each of our sides into direction vectors

ab=b-a=(3 - 7, 1 - 0) = (-4, 1)

bc=c-b=(1,6)

cd=d-c=(4,-1)

da=a-d=(-1,-6)

we see how the opposite sides have negated direction vectors; this is characteristic of a parallelogram.   we have congruent sides ab=cd and bc=da with unequal lengths because 1^2+6^2 is different from (-4)^2+1^2.   so it's not a square or a rhombus.  

we check the dot product of direction vectors of adjacent sides.   we have

ab dot bc = -4(1) + 1(6) = 2

that's not zero so the sides are not perpendicular.   we've ruled out rectangle.

the direction vectors indicate two pairs of congruent parallel sides, so we choose parallelogram over trapezoid.