# Equations 18023

Solve a system of equations with four unknowns:

2a + 2b-c + d = 4
4a + 3b-c + 2d = 6
8a + 5b-3c + 4d = 12
3a + 3b-2c + 2d = 6

a =  1
b =  1
c =  -1
d =  -1

### Step-by-step explanation:

2a+2b-c+d=4
4a+3b-c+2d=6
8a+5b-3c+4d=12
3a+3b-2c+2d=6

2·a+2·b-c+d=4
4·a+3·b-c+2·d=6
8·a+5·b-3·c+4·d=12
3·a+3·b-2·c+2·d=6

2a+2b-c+d = 4
4a+3b-c+2d = 6
8a+5b-3c+4d = 12
3a+3b-2c+2d = 6

Pivot: Row 1 ↔ Row 3
8a+5b-3c+4d = 12
4a+3b-c+2d = 6
2a+2b-c+d = 4
3a+3b-2c+2d = 6

Row 2 - 4/8 · Row 1 → Row 2
8a+5b-3c+4d = 12
0.5b+0.5c = 0
2a+2b-c+d = 4
3a+3b-2c+2d = 6

Row 3 - 2/8 · Row 1 → Row 3
8a+5b-3c+4d = 12
0.5b+0.5c = 0
0.75b-0.25c = 1
3a+3b-2c+2d = 6

Row 4 - 3/8 · Row 1 → Row 4
8a+5b-3c+4d = 12
0.5b+0.5c = 0
0.75b-0.25c = 1
1.125b-0.875c+0.5d = 1.5

Pivot: Row 2 ↔ Row 4
8a+5b-3c+4d = 12
1.125b-0.875c+0.5d = 1.5
0.75b-0.25c = 1
0.5b+0.5c = 0

Row 3 - 0.75/1.125 · Row 2 → Row 3
8a+5b-3c+4d = 12
1.125b-0.875c+0.5d = 1.5
0.3333c-0.3333d = 0
0.5b+0.5c = 0

Row 4 - 0.5/1.125 · Row 2 → Row 4
8a+5b-3c+4d = 12
1.125b-0.875c+0.5d = 1.5
0.3333c-0.3333d = 0
0.8889c-0.2222d = -0.6667

Pivot: Row 3 ↔ Row 4
8a+5b-3c+4d = 12
1.125b-0.875c+0.5d = 1.5
0.8889c-0.2222d = -0.6667
0.3333c-0.3333d = 0

Row 4 - 0.33333333/0.88888889 · Row 3 → Row 4
8a+5b-3c+4d = 12
1.125b-0.875c+0.5d = 1.5
0.8889c-0.2222d = -0.6667
-0.25d = 0.25

d = 0.25/-0.25 = -1
c = -0.66666667+0.22222222222222d/0.88888889 = -0.66666667+0.22222222 · (-1)/0.88888889 = -1
b = 1.5+0.875c-0.5d/1.125 = 1.5+0.875 · (-1)-0.5 · (-1)/1.125 = 1
a = 12-5b+3c-4d/8 = 12-5 · 1+3 · (-1)-4 · (-1)/8 = 1

a = 1
b = 1
c = -1
d = -1

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