Pedestrian up-down hill

Pedestrian goes for a walk first at the plane at 4 km/h, then uphill at 3 km/h. Then it is in the middle of the route, turns back, and goes downhill at 6 km/h. The total walk was 6 hours. How many kilometers went pedestrians?

s =  24 km

Step-by-step explanation:

6 = a+b+c+d
s = 4a+3b+6c+4d
4a=4d
3b=6c
a=0

6 = a+b+c+d
s = 4·a+3·b+6·c+4·d
4·a=4·d
3·b=6·c
a=0

a+b+c+d = 6
4a+3b+6c+4d-s = 0
4a-4d = 0
3b-6c = 0
a = 0

Pivot: Row 1 ↔ Row 2
4a+3b+6c+4d-s = 0
a+b+c+d = 6
4a-4d = 0
3b-6c = 0
a = 0

Row 2 - 1/4 · Row 1 → Row 2
4a+3b+6c+4d-s = 0
0.25b-0.5c+0.25s = 6
4a-4d = 0
3b-6c = 0
a = 0

Row 3 - Row 1 → Row 3
4a+3b+6c+4d-s = 0
0.25b-0.5c+0.25s = 6
-3b-6c-8d+s = 0
3b-6c = 0
a = 0

Row 5 - 1/4 · Row 1 → Row 5
4a+3b+6c+4d-s = 0
0.25b-0.5c+0.25s = 6
-3b-6c-8d+s = 0
3b-6c = 0
-0.75b-1.5c-d+0.25s = 0

Pivot: Row 2 ↔ Row 3
4a+3b+6c+4d-s = 0
-3b-6c-8d+s = 0
0.25b-0.5c+0.25s = 6
3b-6c = 0
-0.75b-1.5c-d+0.25s = 0

Row 3 - 0.25/-3 · Row 2 → Row 3
4a+3b+6c+4d-s = 0
-3b-6c-8d+s = 0
-c-0.66667d+0.33333s = 6
3b-6c = 0
-0.75b-1.5c-d+0.25s = 0

Row 4 + Row 2 → Row 4
4a+3b+6c+4d-s = 0
-3b-6c-8d+s = 0
-c-0.66667d+0.33333s = 6
-12c-8d+s = 0
-0.75b-1.5c-d+0.25s = 0

Row 5 - -0.75/-3 · Row 2 → Row 5
4a+3b+6c+4d-s = 0
-3b-6c-8d+s = 0
-c-0.66667d+0.33333s = 6
-12c-8d+s = 0
d = 0

Pivot: Row 3 ↔ Row 4
4a+3b+6c+4d-s = 0
-3b-6c-8d+s = 0
-12c-8d+s = 0
-c-0.66667d+0.33333s = 6
d = 0

Row 4 - -1/-12 · Row 3 → Row 4
4a+3b+6c+4d-s = 0
-3b-6c-8d+s = 0
-12c-8d+s = 0
0.25s = 6
d = 0

Pivot: Row 4 ↔ Row 5
4a+3b+6c+4d-s = 0
-3b-6c-8d+s = 0
-12c-8d+s = 0
d = 0
0.25s = 6

s = 6/0.25 = 24
d = 0/1 = 0
c = 0+8d-s/-12 = 0-24/-12 = 2
b = 0+6c+8d-s/-3 = 0+6 · 2-24/-3 = 4
a = 0-3b-6c-4d+s/4 = 0-3 · 4-6 · 2+24/4 = 0

a = 0
b = 4
c = 2
d = 0
s = 24

Our linear equations calculator calculates it.

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