Barbara's Responses in WyzAnt Answers

2ab + b^2/b =        

2ab + b*b/b =                 Definition of exponent

2ab + 1b =                      Identity property (b/b = 1)

b(2a +1)  Simplified form.  Use distributive property

Let x=# bullseyes

     y=# 2-point shots

     z=# missed shots

Solve so that both of the following equations are TRUE

#1 x + y + z = 50               where x, y, z are positive integers

#2 7x + 2y + 0z = 152        where x, y, z are positive integers

What is needed are all points (x,y,z) that meet the above criteria.

Maximum values:

x:     7x <= 152                 where x is a positive EVEN number

          x <= 21                  why? because of the property of integers; an odd + even integer = odd

                                        And odd x odd = odd, but 7x must be even in order for 7x+2y+0z = 152.

y:      2y <= 152                 where y is a positive number

           y <= 76                   but there’s an upward limit of 50!

           y <= 50                   because of #1 equation x + y + z = 50

Now let’s remove z as a variable from #1 and #2 above.

#1a    x + y <= 50

#2a  7x + 2y = 152 (subtract 1a from 2a)

       ------------------

        6x + y <= 102

               y <= -6x + 102

Substitute into #1a

       x + -6x + 102 <= 50

                       52 <= 5x

                    10.2 <= x <= 21    where x is a positive EVEN number (remember max above?)

valid values of x are {12, 14, 16, 18, 20}

Solve #2a for y.

        7x + 2y = 152

               2y = 152 + -7x

                 y = 76 + 7x/2

now find y as f(x) = -76 + 7x/2 for all valid values of x

              f(12) = -76 + 42 = 34

              f(14) = 27

              f(16) = 20

              f(18) = 13

              f(20) = 6

valid values of y are {34, 27, 20, 13, 6}

Now find z as f(x,y) = 50 – x – y (#1 solved for z)

             f(12,34) = 4

             f(14,27) = 9

             f(16,20) = 14

             f(18,13) = 19

              f(20,6) = 24

So there are 5 answers:

        (12,34,4)       12 bullseyes, 34 2-pointers, 4 misses

        (14,27,9)       14 bullseyes, 27 2-pointers, 9 misses

        (16,20,14)     16 bullseyes, 20 2-pointers, 14 misses

        (18,13,19)     18 bullseyes, 13 2-pointers, 19 misses

        (20,6,24)       20 bullseyes, 6 2-pointers, 24 misses

Use these terms and do an Internet seach for the examples.

TERM                 ANGLE DEFINITION

Included             The angle made by two lines with a common vertex.
                         This term is the one that includes all the following
                         angles terms. It is the superset.

Angles from a protractor (for a circle):
----------------------------------------
acute                  greater than 0 degrees and less than 90 degrees
right                   90 degrees
obtuse                greater than 90 degrees and less than 180 degrees
straight               180 degrees (vertex attaches 2 half lines going opposite directions)
reflex                  greater than 180 and less than 360 degrees
full rotation          360 degrees (vertex attaches 2 half lines that are the same)

Angles from two intersecting lines:
-----------------------------------
adjacent              formed when two intersecting lines form angles
                          that are supplimentary (side by side) to each other.
                          The sum of the degrees when both angles are added = 180.
                          They share a vertex, share the same two lines,
                           and share one common arm (portion) on those lines.
                           vertical formed when two intersecting lines form angles that
                           share a common vertex, share the same two lines,
                           but share no common arms (portions) of those lines.

opposite               same definition as vertical. They share the same vertex.

When angles are created by a transversal line intersecting with two
parallel lines, we also get these angles:
-------------------------------------------------------------------
corresponding       angles with the same degree on both parallel lines,
                           on the SAME side of the transveral line
alternating            angles with the same degree on both parallel lines,
                           on the OPPOSITE side of the transversal line.

Angles in an isosceles triangle:
--------------------------------
vertex                  angle formed by the two congruent (equal) sides
base                    angle formed by the base (non equal) side
                           note: the other side is one of the congruent sides.

Angles inside and outside a polygon:
------------------------------------
interior                 angles created inside the polygon shape,
                           formed by two sides and the vertex attaching them
exterior                angles formed by an adjacent angle (see above) for
                           the sides that reside on a line.

An equation that includes a radical (squareroot) would have more than one solution. 

Sqrt 4 has two solutions, 2 (e.g. 2*2=4) and -2 (e.g. -2*-2=4).

Use Sqrt in a division expression.   Sqrt 4 / 2 = ?   The answers are 1 and -1.

You could be creative and have two SQRTs.  Sqrt 4 / Sqrt 4 = ?.  The answers are still 1 and -1.

In the answer provided by Cornelia M., it is assumed that p=x and c=y.  You would label the x axis "p" and the y axis "c".  Point (0,24) would have p=0 and c=24.  Point (30,0) would have p=30 and c=0.

This is the "solution" to the equation, all points in the shaded area, not including points directly on the dashed line.