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?
+876 $3(w+x+y+z) = 25$
$w + x + y + z = \frac{25}{3}$
$z = \frac{27}{3} = 9$
$y = \frac{29}{3}$
$x = - \frac{32}{3}$
$w = - \frac{1}{2}$
$\left(\left(- \frac{32}{3}\right) \cdot \left(- \frac{1}{2}\right) \right) + \left(\frac{29}{3} \cdot 9 \right) = \frac{16}{3} + \frac{261}{3} = \boxed{\frac{277}{3}}$
+240 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
+876 Apparently none of us are correct, I just checked with WA https://www.wolframalpha.com/input/?i=w%2Bx%2By+%3D+-2++w%2Bx%2Bz+%3D+-4++w%2By%2Bz+%3D+19++x%2By%2Bz+%3D+12%2C, and then https://www.wolframalpha.com/input/?i=%28228%2F49*+-48%2F49%29+%2B+%28408%2F49+*+156%2F7%29. The answer is $\frac{434592}{2401}$
+240 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
+129850 w + x + y = -2 (1)
w + x + z = -4 (2)
w + y + z = 19 (3)
x + y + z = 12 (4)
Add the first three equations
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)
Add (5) and (6)
y - z = 2 (7)
And manipulating (3)
y + z = 19 + 11/3 ⇒ y + z = 68/3 (8)
Add (7) and (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
+1641 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
