All Answers 355923 Answers

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

Great! That helps a lot! Thank you so much for your help! Have a wonderful day/night!

]Suppose $f(x)$ is a function that has this property:  For all real numbers $a$ and $b$ such that $a

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  !!!

Thank you so much!

I think I worked that last two out but I'm not confident in my answers. The answer I got for 47 is 1494 ft and the answer that I got for 51 is 1881 mi. Are these correct? Sorry.

But I have to do these now. Am I supposed to get c as well? I attempted #10 but I didn't do it well lol so I just erased everything. Dang it I have to go soon but want to get this done today!

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...!!!!!

x= 2

y=3

(2,3)

Thanks I have 2 more that are a bit different cuz they involve 3 each and also a,b, and c. Would you be able to help with that seperately?

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...!!!

I knew I wouldn't have gotten it correct -_-

I tried 8 but it's obviously completely wrong. *sigh*

x+7y=23

-7y    -7y

x=23-7y

8(23-7y)-3y=7

184-56y-3y=7

and here is where it all goes wrong...

184-53y=7

-184      -184

53y=-177

/53     /53

y=-3.34 or something like that >.<

and i'll wait to see if i'm even correct first

Try 5  and note that     3x + y  = -2  ⇒   y  = -2- 3x

Yes you can...you're close  !!!

Good try  !!!...  you have the right idea..just a small error

Let's take it from  here....

4x+3(7-7x)=21

4x + 21 - 21x  =  21     subtract  21  from both sides

4x  - 21x   =0

-17x  = 0   ⇒    x  = 0

Sub this back into y  = 7 - 7x  →  y  = 7 - 7 (0)    = 7  - 0   = 7

(x,y)  =(0,7) ...check this solution in both equations to see that this is true....!!!!

|x - 4| + |x + 4| <= 10

We have these  equations

x - 4  + x + 4   <=  10

2x  <=  10

 x  <= 5

And 

x - 4  + x + 4   >=  -10

2x   >= -10

x >= -5

So.....the solution  is   [-5 , 5  ]

Here's the graph of the solution  :  https://www.desmos.com/calculator/7ar4jqmtg3

Oh wait I can try 7! 7x+y=7 ( subtract 7x from both sides)

y=7-7x

4x+3(7-7x)=21

(21-21x)

4x+21-21x=21

25x+21=21

-21        -21

25x do I divide it by 25 to get 1?

x=1

y=7-7(1)

y=7-7

y=0

(x,y)-->(1,0)

Yep... D  too   1/16  = .0625

My brain is about to explode, I can't do this >.<

You're not wrong...but  the there is an easier way without resorting to  those nasty  fractions

6)

x + 4y  = --6

8x  - 2y = -14

Do you see that we have a  single  variable in the first equation ???...in this case, x

So   subtract  4y  from  both  sides  and we get   x   = -6  - 4y

Now..in the second equatiion....just substitute this  for x  and we have

8 [ -6 - 4y ] - 2y  = -14    simplify

-48 - 32y - 2y  =  -14

-48  - 34y  = -14     add 48 to both sides

-34y = 34   divide both sides by  -34

y = -1

The beauty here is that  we already have an expression for x...we just need to fill in the y value

x = -6 - 4(-1)   =    -6 + 4   =  -2

So  (x, y)  = (  -2, -1)

As a hint on number 7...note that  7x  + y  = 7  can be transformed into   y = 7 - 7x

Can you take it from here???...post your work if you get stuck...I'll see if I can "un-stick" you  !!...LOL!!!!

Do D doesnt work?

4^3*4^-5=4^3+-5=4^-2, so (A), and 4^-2=1/16, and that is also (B). (H) also works two: 1/4^2=1/16. (I) also works! and, (j) too, 4^3/4^5=1/4^2=1/16. 

Okay um -2y=-14+8x divide by -2 on both sides so y=-14+8x/-2

4y=-6x divide by 4 on both sides as well and you get y=-6x/4

-6x/4=-14+8x/-2

So far so good? or am i totally wrong?