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# system

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If w, x, y, and z are real numbers satisfying:

w+x+y = -2

w+x+z = -4

w+y+z = 19

x+y+z = 12,

what is wx + yz?

Jun 30, 2021

#1
-1

$3(w+x+y+z) = 25$

$w + x + y + z = \frac{25}{3}$

$z = \frac{27}{3} = 9$

$y = \frac{29}{3}$

$x = - \frac{32}{3}$

$w = - \frac{1}{2}$

$\left(\left(- \frac{32}{3}\right) \cdot \left(- \frac{1}{2}\right) \right) + \left(\frac{29}{3} \cdot 9 \right) = \frac{16}{3} + \frac{261}{3} = \boxed{\frac{277}{3}}$

Jun 30, 2021
#2
-1

Add all the four equations together to get,

$$3(w+x+y+z)=25$$

$$w+x+y+z=8\frac{1}{3}$$

Subtract it from each of the equations to get,

$$z = 10\frac{1}{3}$$

$$y = 12\frac{1}{3}$$

$$x = -10\frac{2}{3}$$

$$w = -3\frac{2}{3}$$

Solve for the question,

$$30\frac{4}{9} + 120\frac{1}{9} = 150\frac{5}{9}$$

So $$150\frac{5}{9}$$ is the final answer

Jun 30, 2021
#4
0

Seems too complex of an answer?

Edit: You misplaced commas: https://www.wolframalpha.com/input/?i=w%2Bx%2By+%3D+-2%2C++w%2Bx%2Bz+%3D+-4%2C++w%2By%2Bz+%3D+19%2C+x%2By%2Bz+%3D+12%2C

Awesomeguy  Jun 30, 2021
edited by Awesomeguy  Jun 30, 2021
#5
0

Yea... wonder who made this problem!

MathProblemSolver101  Jun 30, 2021
#6
+2

w  +  x  +  y  =   -2        (1)

w + x +   z  =   -4           (2)

w  +  y  +  z  =   19        (3)

x + y +   z  = 12             (4)

3w  +  2 ( x  +  y  + z)  =  13

3w  +  2 ( 12)  =    13

3w  +  24   =  13

3w   =  -11

w   =   -11/3

Manipualting  the  first two equations

x +  y =  -2  + 11/3    ⇒   x +  y  =   5/3            (5)

x + z =  -4  +   11/3   ⇒   x  +  z  =  -1/3  ⇒    -x  - z  =  1/3       (6)

y - z  =  2          (7)

And  manipulating (3)

y +  z  =  19  +  11/3    ⇒       y + z =   68/3         (8)

2y   =  74/3

y  = 74/6   =  37/3

And

z    =  68/3 - 37/3  =  31/3

And

x =  5/3 - 37/3   =   -32/3

wx  + yz   =

(-11/3)(-32/3)   +  (37/3)  ( 31/3)     =

1499  /  9   Jun 30, 2021
#7
+5

w + x + y = -2

w + x + z = -4

w + y + z = 19

x + y + z = 12

~~~~~~~~~~~~~~~~~

w + x + z = -4

w + y + z = 19

y = x + 23                     y = 12 1/3

~~~~~~~~~~~~~~~~~~

w + y + z = 19

x + y + z = 12

w = x + 7                      w = - 3 2/3

~~~~~~~~~~~~~~~~~~~

w + x + y = -2

(x + 7) + x + (x + 23) = -2

x = - 10 2/3

~~~~~~~~~~~~~~~~~~~

x + y + z = 12

z = 12 - (-10 2/3) - (12 1/3)

z = 10 1/3

~~~~~~~~~~~~~~~~~~~~

w*x + y*z = 166 5/9

Jul 1, 2021
edited by jugoslav  Jul 1, 2021