# Determine 3033

Andrej, Michal, Patrik, and Juraj together weigh 128 kg. Their weights are in the ratio 1:6:7:2. Determine what weight each of them has.

a =  8
j =  16
m =  48
p =  56

### Step-by-step explanation:

a+m+p+j=128
m = 6a
p = 7a
j=2a

a+m+p+j=128
m = 6·a
p = 7·a
j=2·a

a+j+m+p = 128
6a-m = 0
7a-p = 0
2a-j = 0

Pivot: Row 1 ↔ Row 3
7a-p = 0
6a-m = 0
a+j+m+p = 128
2a-j = 0

Row 2 - 6/7 · Row 1 → Row 2
7a-p = 0
-m+0.8571p = 0
a+j+m+p = 128
2a-j = 0

Row 3 - 1/7 · Row 1 → Row 3
7a-p = 0
-m+0.8571p = 0
j+m+1.1429p = 128
2a-j = 0

Row 4 - 2/7 · Row 1 → Row 4
7a-p = 0
-m+0.8571p = 0
j+m+1.1429p = 128
-j+0.2857p = 0

Pivot: Row 2 ↔ Row 3
7a-p = 0
j+m+1.1429p = 128
-m+0.8571p = 0
-j+0.2857p = 0

Row 4 + Row 2 → Row 4
7a-p = 0
j+m+1.1429p = 128
-m+0.8571p = 0
m+1.4286p = 128

Row 4 + Row 3 → Row 4
7a-p = 0
j+m+1.1429p = 128
-m+0.8571p = 0
2.2857p = 128

p = 128/2.28571429 = 56
m = 0-0.85714285714286p/-1 = 0-0.85714286 · 56/-1 = 48
j = 128-m-1.1428571428571p = 128-48-1.14285714 · 56 = 16
a = 0+p/7 = 0+56/7 = 8

a = 8
j = 16
m = 48
p = 56

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