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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 5, 2022

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Solve the following system:
{w + x + y = -2 | (equation 1)
w + x + z = -4 | (equation 2)
w + y + z = 19 | (equation 3)
x + y + z = 12 | (equation 4)

Subtract equation 1 from equation 2:
{w + x + y + 0 z = -2 | (equation 1)
0 w + 0 x - y + z = -2 | (equation 2)
w + 0 x + y + z = 19 | (equation 3)
0 w + x + y + z = 12 | (equation 4)

Subtract equation 1 from equation 3:
{w + x + y + 0 z = -2 | (equation 1)
0 w + 0 x - y + z = -2 | (equation 2)
0 w - x + 0 y + z = 21 | (equation 3)
0 w + x + y + z = 12 | (equation 4)

Swap equation 2 with equation 3:
{w + x + y + 0 z = -2 | (equation 1)
0 w - x + 0 y + z = 21 | (equation 2)
0 w + 0 x - y + z = -2 | (equation 3)
0 w + x + y + z = 12 | (equation 4)

Add equation 2 to equation 4:
{w + x + y + 0 z = -2 | (equation 1)
0 w - x + 0 y + z = 21 | (equation 2)
0 w + 0 x - y + z = -2 | (equation 3)
0 w + 0 x + y + 2 z = 33 | (equation 4)

Add equation 3 to equation 4:
{w + x + y + 0 z = -2 | (equation 1)
0 w - x + 0 y + z = 21 | (equation 2)
0 w + 0 x - y + z = -2 | (equation 3)
0 w + 0 x + 0 y + 3 z = 31 | (equation 4)

Divide equation 4 by 3:
{w + x + y + 0 z = -2 | (equation 1)
0 w - x + 0 y + z = 21 | (equation 2)
0 w + 0 x - y + z = -2 | (equation 3)
0 w + 0 x + 0 y + z = 31/3 | (equation 4)

Subtract equation 4 from equation 3:
{w + x + y + 0 z = -2 | (equation 1)
0 w - x + 0 y + z = 21 | (equation 2)
0 w + 0 x - y + 0 z = -37/3 | (equation 3)
0 w + 0 x + 0 y + z = 31/3 | (equation 4)

Multiply equation 3 by -1:
{w + x + y + 0 z = -2 | (equation 1)
0 w - x + 0 y + z = 21 | (equation 2)
0 w + 0 x + y + 0 z = 37/3 | (equation 3)
0 w + 0 x + 0 y + z = 31/3 | (equation 4)

Subtract equation 4 from equation 2:
{w + x + y + 0 z = -2 | (equation 1)
0 w - x + 0 y + 0 z = 32/3 | (equation 2)
0 w + 0 x + y + 0 z = 37/3 | (equation 3)
0 w + 0 x + 0 y + z = 31/3 | (equation 4)

Multiply equation 2 by -1:
{w + x + y + 0 z = -2 | (equation 1)
0 w + x + 0 y + 0 z = -32/3 | (equation 2)
0 w + 0 x + y + 0 z = 37/3 | (equation 3)
0 w + 0 x + 0 y + z = 31/3 | (equation 4)

Subtract equation 2 from equation 1:
{w + 0 x + y + 0 z = 26/3 | (equation 1)
0 w + x + 0 y + 0 z = -32/3 | (equation 2)
0 w + 0 x + y + 0 z = 37/3 | (equation 3)
0 w + 0 x + 0 y + z = 31/3 | (equation 4)

Subtract equation 3 from equation 1:
{w + 0 x + 0 y + 0 z = -11/3 | (equation 1)
0 w + x + 0 y + 0 z = -32/3 | (equation 2)
0 w + 0 x + y + 0 z = 37/3 | (equation 3)
0 w + 0 x + 0 y + z = 31/3 | (equation 4)

w = -11/3

x = -32/3
y = 37/3
z = 31/3           wx + yz ==[-11/3*-32/3] + [37/3*31/3]==1,499 / 9 ==166 5/9

Jun 5, 2022