CPhill

It wouldn't change the answer, much....the graph of  x + y ≤ 4  would be a solid line instead of the dashed line of  x + y < 4

The answer for the chosen points would be the same...

OK...easy enough....

Here's the desmos graph : https://www.desmos.com/calculator/h3ywx9slgp

Note  that the point  ( 0,0)  is a solution because   0 > -2   and  0 + 0  < 4

Note that the point  (0,5)  is not a solution  because  5 > -2   but   0 + 5  is  not less than 4

No numbers are showing up....  

c =  4  in the first one

See (10)  below...

10)

2a  - 4b  - 2c  =   -4       divide by 2  ⇒   a  - 2b  - c   =  -2    (1)

-3a   + 3b  +6c   = 30    divide by 3  ⇒  - a  + b + 2c  = 10    (2)

-a  - 3b  + c   = 17                                  -a  - 3b  + c  = 17     (3)

Add   (1)  and (2)                               Add  (1) and (3)

a  - 2b  -  c  =  -2                                -a - 2b  - c   = -2   

-a + b  + 2c   = 10                               a   -3b + c  = 17

________________                          ______________

       -b  + c  = 8 ..so..                                -5b    =  15

        c  =  8 + b   (4)                                      b  = -3    (5)

 Sub (5)  into (4)     

c = 8 + (-3)   

c  =  5

And using  (1)  to find  a

a  - 2b  - c   =  -2

a  - 2(-3) - (5)    = -2

a  + 6 - 5  = -2

a + 1   = -2

a   =  -3

47) A pentagonal prism 6 ft tall with a regular base measuring 12 ft on each edge and an apothem of length 8.3 ft.

We will have  5congruent triangles forming the base...the height is 8.3 ft and the base =  12 ft 

So...the base area  is  5*  (1/2) (8.3) * ( 12) =  249  ft^2

And the volume is  base area x height  =   249 * 6  =   1494  ft^3    CORRECT  !!!

So  51 is similar....the volume is  9 *  5(1/2)(11) ( 7.6)  =   1881  (miles??)^3....that's a  big prism!!!!...LOL!!!

If you're  still onboard...let me show you how to  solve these with something  known as  Cramer's Rule

Here's  6...which we know the answer to

x + 4y   = -6

8x  - 2y  = -14   we have the equations lined up properly....

Now  write the left side just using the coefficients

1    4

8   -2        [ multiply  (1 * -2) diagonally  = -2  .....  subtract the product of the other diagonal =  (  8 * 4) = 32 ]

So we have    -2 -(32)   = -34       (1)         [as long as this isn't  0, we're  Ok]

Sub  the  -6  and  -14  into the 1st column  of  (1) ...this will  be the  "x" column"

-6    4      

-14  -2   [ repeat the process we just did...  (-6 * -2)   - ( 4 * -14)  = ( 12  - (-56) )  =    68

Divide this result by  the result  from (1)  and we get     68 / -34   =  -2   = x

Now  sub   -6 and -14 into the second column of (1)...this will be the "y" column ... and repeat the process

1   -6

8    -14          (1 * -14)  - ( -6 * 8)   =   ( -14 )  - ( -48)   =  -14  +  48  = 34

Divide this result  by (1)  to  get y   =   34/ -34  =  -1

So  (x, y) =   ( -2, -1)   ....exactly what we had before  !!!

32) A prism 7 cm tall with a right triangle for a base with side lengths 6 cm, 8 cm, and 10 cm.

You may not recognize this...but the base forms a "Pythagorean"  right triangle  with legs  of  6 and 8  cm

The area of a right triangle is  (1/2)  ( product of leg lengths)

So...the area of the base   = (1/2) (6 * 8 ) = (1/2) 48  = 24 cm^2

And the volume of the prism is just the   base area  x  height =   24  x  7  =   192 cm^2

These are difficult....and often lengthy......the other problem is that there is no "set" way to  proceed.....[P.S.....watch your math carefully....one mistake dooms you  !!! ]

I'll do the first one  just to demonstrate...

2a  + 3b  - 5c   = -5    (1)

-2a - 2b  - c  = -14      (2)

4a  - 4b   + 4c  = -4       divide this equation by 2   ⇒  2a  - 2b  + 2c = -2   (3)

Add  (1)  and (2)                               Add (2)  and (3)

2a + 3b - 5c  = -5                              -2a   - 2b  - c  =  -14

-2a - 2b - c  = -14                               2a   - 2b  + 2c  =  -2

________________                         _________________

      b  - 6c    =  -19    (4)                             -4b  + c  = -16   ⇒   c  =   4b   - 16      (5)

Sub  (5)  into ( 4)  and we have

b - 6(4b - 16) =  -19

b - 24b + 96  = -19

-23b  + 96  = -19   subtract 96 from both sides

-23b = -115    divide both sides by  -23

b  = 5 

Use  (5)   to  find  c

c = 4(5) - 16  =   20  - 16 =  4

And using  (3) to  find  "a"  we have

2a  - 2(5)  + 2(4) = -2

2a  - 10  + 8  = -2

2a -2 = -2   add 2 to both sides

2a  = 0 ⇒     a   = 0 

RP...this requires a bit of experience to master...I would stick to  solving equations in two  variables and make sure you felt comfortable with that before  you attmpted these.....just  my opinion....

As a disclaimer...I can show you something known as "Cramer's  Rule"  where no solving is involved...however...it's also  a little  lengthy...but not difficult...!!!!!

184-56y-3y=7

and here is where it all goes wrong...

LOL!!!!!.....Just a math error....

184 - 59y  = 7    subtract 184 from both sids

-59y  = -177  divide both sides by  -59

y  = 3

Sub this back into   x=23-7y   to  find x...and you're finished  !!!

You made a really good try...!!!