Luggage and air travel

Two friends traveling by plane had a total of 35 kg of luggage. They paid one 72 CZK and the second 108 CZK for being overweight. If only one paid for all the bags, it would cost 300 CZK. What weight of baggage did each of them have? How many kilograms of luggage could a single passenger carry for free?

Correct answer:

a = 16 kg
b = 19 kg
x = 10 kg

Step-by-step explanation:

a+b = 35
a-x = 72p
b-x = 108p
a+b-x = 300p

a+b = 35
a-x = 72·p
b-x = 108·p
a+b-x = 300·p

a+b = 35
a-72p-x = 0
b-108p-x = 0
a+b-300p-x = 0

Row 2 - Row 1 → Row 2
a+b = 35
-b-72p-x = -35
b-108p-x = 0
a+b-300p-x = 0

Row 4 - Row 1 → Row 4
a+b = 35
-b-72p-x = -35
b-108p-x = 0
-300p-x = -35

Row 3 + Row 2 → Row 3
a+b = 35
-b-72p-x = -35
-180p-2x = -35
-300p-x = -35

Pivot: Row 3 ↔ Row 4
a+b = 35
-b-72p-x = -35
-300p-x = -35
-180p-2x = -35

Row 4 - -180/-300 · Row 3 → Row 4
a+b = 35
-b-72p-x = -35
-300p-x = -35
-1.4x = -14

x = -14/-1.4 = 10
p = -35+x/-300 = -35+10/-300 = 0.08333333
b = -35+72p+x/-1 = -35+72 · 0.08333333+10/-1 = 19
a = 35-b = 35-19 = 16

a = 16
b = 19
p = 1/12 ≈ 0.083333
x = 10

Our linear equations calculator calculates it.