Gold rods

In one kingdom, two gold rods were inherited over generations. But King Emanuel had three sons who could not agree on anything. He therefore wanted to break one rod so as to make three rods out of two. The youngest son gets the shortest rod, the oldest son gets the longest rod. The middle son gets a rod which is shorter than the longest rod by as much as it is longer than the shortest rod.
What lengths of rods will the princes get, if both original rods had a length of 78 cm?

Final Answer:

a =  26 cm
b =  52 cm
c =  78 cm

Step-by-step explanation:

a+b+c = 2·78
c = 78
b+x = c
b-x = a

a+b+c = 156
c = 78
b-c+x = 0
a-b+x = 0

Row 4 - Row 1 → Row 4
a+b+c = 156
c = 78
b-c+x = 0
-2b-c+x = -156

Pivot: Row 2 ↔ Row 4
a+b+c = 156
-2b-c+x = -156
b-c+x = 0
c = 78

Row 3 - 1/-2 · Row 2 → Row 3
a+b+c = 156
-2b-c+x = -156
-1.5c+1.5x = -78
c = 78

Row 4 - 1/-1.5 · Row 3 → Row 4
a+b+c = 156
-2b-c+x = -156
-1.5c+1.5x = -78
x = 26


x = 26/1 = 26
c = -78-1.5x/-1.5 = -78-1.5 · 26/-1.5 = 78
b = -156+c-x/-2 = -156+78-26/-2 = 52
a = 156-b-c = 156-52-78 = 26

a = 26
b = 52
c = 78
x = 26

Our linear equations calculator calculates it.

$b=52=52\text{cm}$

$c=78=78\text{cm}$

Try another example