CPhill

https://en.wikipedia.org/wiki/Leonhard_Euler

If m∠A + m∠B = m∠B + m∠C, subtract  m∠B from both sides  [ subtraction property of equality ].

We get that m∠A  = m∠C     and by the  symmetric property of equality,  m∠C = m∠A

x + y  = 6     (1)

x = y + 2  subtract y from  both sides ⇒  x - y = 2   (2)

Add (1)  and (2) ⇒   2x  = 8  ⇒   divide both sides by 2 ⇒  x  = 4

Sub this answer  into (1)  [ we could use 2, as well ]

x + y  = 6

4 + y  = 6

So...y   must = 2

(4, 2)

x  = (1/2) y + 5      multiply through by 2 to clear the fraction ⇒  2x  = y + 10    (1)    

2x + 3y  = -14     (2)

Sub (1) into (2)

[ y + 10 ] + 3y  = -14

4y + 10  = -14      subtract 10  from  both  sides

4y = -24    divide through by  4

y  = - 6

Put this into (1) for y ⇒  2x = -6 + 10  ⇒  2x  = 4  ⇒ divide both sides by  2 ⇒  x = 2

So.... ( 2, -6)  is the solution

3x + y  = 4     (1)

2x + y  = 7       multiply this through  by -1 ⇒  -2x - y  = -7   (2)

Add (1) and (2)   ⇒  x  = -3

Subbing into  either equation to  find y  we get  3 (-3) + y  = 4 ⇒  -9 + y  = 4  add 9 to  both sides  y   = 13

(-3, 13)  is the solution 

-2x + 6y  = -38       multiply  through by  3 ⇒   -6x + 18y  = -114    (1)

3x - 4y  = 32           multiply through  by 2 ⇒   6x - 8y  = 64   (2)

Add (1) and (2) ⇒  10y  = -50    ⇒ divide through by 10 ⇒  y  = - 5

Subbing this back into the first equation to find  x, we get  -2x + 6(-5)  = -38   

-2x - 30  = -38

Add 30 to   both sides  -2x  = -8

Divide both sides by -2

x  = 4

(4, -5)  is the solution     

Last one

x + 3y - z  = 6     (1)

4x - 2y + 2z  = -10    divide through by 2 ⇒    2x - y + z  = -5       (2)

6x + z  = -12  ⇒    z  = -12 - 6x   (3)

Sub (3)  into (1) and  (2)  for  z

x + 3y - [ -12 - 6x] = 6 ⇒    x + 3y + 12 + 6x  = 6 ⇒  7x + 3y  = -6    (4)

2x - y + [ -12 - 6x] = -5 ⇒   2x - y - 12 - 6x  = -5  ⇒ -4x - y = 7  mulltiply through by 3 ⇒

-12x - 3y  = 21  ( 5)

Add (4)  and (5)

-5x  = 15      divide through by -5

x  = -3

Using (4)  we can  find y ⇒  7(-3) + 3y  = -6  ⇒  -21 + 3y = -6 ⇒  3y = 15 ⇒ y = 5

Using (3) we can find z ⇒  z = -12 -6(-3) ⇒  z = -12 + 18 ⇒  z  = 6

So...the solution is  (-3, 5, 6)

No prob....!!!!

|x^2 - 14x + 29| = 4

We have that  either

x^2 - 14x + 29   = 4          or that         x^2  - 14x  +  29   = -4           simplify

x^2 - 14x  + 25  = 0                            x^2 - 14x + 33  = 0               factor

The second equation   factors as  ( x - 11) ( x - 3)  = 0

Setting  both factors to 0  and solving for x produces   x = 11   and x  = 3

For the first equation we can complete the square on x  to solve

x^2  - 14x   = -25       take 1/2 of 14  = 7....square it  = 49....add it to both sides

x^2 - 14x + 49 = -25 + 49       factor the left side....simpify the right

(x - 7)^2  = 24       take  both roots

x - 7  = ±√24

x - 7 = ±2√6    add 7 to  both sides

x = 7 ±2√6

None of the 4 solutions are extraneous   as we  can see by this graph....the x axis is  intersected  four times  [ 4 real "zeroes"]

https://www.desmos.com/calculator/8incrgx666

LOL!!!.....I did have a "personalized" copy, once....but I think Sisyphus absconded with it in his search for the Roman "zero"

We have 26 ways to choose the first and last letters

Coldplay. you don't specify if the intervening letters can  be repeats of the first [and last]  chosen letter....but...if we don't have any restrictions on the intervening letters.....we have  26 choices for the first letter, 26 for the second, 26 for the third...and only 1 for the last  [ whichever letter you chose first ] =  26^3  = 17576 "words"

Thanks...I try to give correct answers, but I don't promise anything !!!

The total area of the target  =  pi * 10^2  = 100 pi in^2

One black area is just the black area of a circle with a radius of 4 inches  = pi*4^2   = 16 pi in^2

The second black area is the area between the 6 in circle and the 8 in  circle =   the difference in areas of a circle with a radius of 8 in  and a circle with a radius of 6 in  =

pi [ 8^2  - 6^2] = pi [ 64 - 36 ]=    28 pi  in^2

So...P(black area)   =  Black Areas / Total Area =  [ 16 + 28 ] pi /  [ 100 pi]  =  44/100  = 11/25