The ages

The ages of the four sons make an arithmetic sequence, the sum of which is the father's age today. In three years, the father's age will be the sum of the ages of the three eldest sons, and in the next two and three months, the father's age will be the sum of the ages of the three youngest sons. What are the ages of all five people today?

a =  6
b =  8
c =  10
d =  12
f =  36
x =  2

Step-by-step explanation:

b=a+x
c = b+x
d = c+x
f = a+b+c+d
f+3 = (b+3)+(c+3)+(d+3)
f+5 = 2+(a+5)+(b+5)+(c+5)

a-b+x = 0
b-c+x = 0
c-d+x = 0
a+b+c+d-f = 0
b+c+d-f = -6
a+b+c-f = -12

Row 4 - Row 1 → Row 4
a-b+x = 0
b-c+x = 0
c-d+x = 0
2b+c+d-f-x = 0
b+c+d-f = -6
a+b+c-f = -12

Row 6 - Row 1 → Row 6
a-b+x = 0
b-c+x = 0
c-d+x = 0
2b+c+d-f-x = 0
b+c+d-f = -6
2b+c-f-x = -12

Pivot: Row 2 ↔ Row 4
a-b+x = 0
2b+c+d-f-x = 0
c-d+x = 0
b-c+x = 0
b+c+d-f = -6
2b+c-f-x = -12

Row 4 - 1/2 · Row 2 → Row 4
a-b+x = 0
2b+c+d-f-x = 0
c-d+x = 0
-1.5c-0.5d+0.5f+1.5x = 0
b+c+d-f = -6
2b+c-f-x = -12

Row 5 - 1/2 · Row 2 → Row 5
a-b+x = 0
2b+c+d-f-x = 0
c-d+x = 0
-1.5c-0.5d+0.5f+1.5x = 0
0.5c+0.5d-0.5f+0.5x = -6
2b+c-f-x = -12

Row 6 - Row 2 → Row 6
a-b+x = 0
2b+c+d-f-x = 0
c-d+x = 0
-1.5c-0.5d+0.5f+1.5x = 0
0.5c+0.5d-0.5f+0.5x = -6
-d = -12

Pivot: Row 3 ↔ Row 4
a-b+x = 0
2b+c+d-f-x = 0
-1.5c-0.5d+0.5f+1.5x = 0
c-d+x = 0
0.5c+0.5d-0.5f+0.5x = -6
-d = -12

Row 4 - 1/-1.5 · Row 3 → Row 4
a-b+x = 0
2b+c+d-f-x = 0
-1.5c-0.5d+0.5f+1.5x = 0
-1.333333d+0.333333f+2x = 0
0.5c+0.5d-0.5f+0.5x = -6
-d = -12

Row 5 - 0.5/-1.5 · Row 3 → Row 5
a-b+x = 0
2b+c+d-f-x = 0
-1.5c-0.5d+0.5f+1.5x = 0
-1.333333d+0.333333f+2x = 0
0.333333d-0.333333f+x = -6
-d = -12

Row 5 - 0.33333333/-1.33333333 · Row 4 → Row 5
a-b+x = 0
2b+c+d-f-x = 0
-1.5c-0.5d+0.5f+1.5x = 0
-1.333333d+0.333333f+2x = 0
-0.25f+1.5x = -6
-d = -12

Row 6 - -1/-1.33333333 · Row 4 → Row 6
a-b+x = 0
2b+c+d-f-x = 0
-1.5c-0.5d+0.5f+1.5x = 0
-1.333333d+0.333333f+2x = 0
-0.25f+1.5x = -6
-0.25f-1.5x = -12

Row 6 - Row 5 → Row 6
a-b+x = 0
2b+c+d-f-x = 0
-1.5c-0.5d+0.5f+1.5x = 0
-1.333333d+0.333333f+2x = 0
-0.25f+1.5x = -6
-3x = -6

x = -6/-3 = 2
f = -6-1.5x/-0.25 = -6-1.5 · 2/-0.25 = 36
d = 0-0.33333333333333f-2x/-1.33333333 = 0-0.33333333 · 36-2 · 2/-1.33333333 = 12
c = 0+0.5d-0.5f-1.5x/-1.5 = 0+0.5 · 12-0.5 · 36-1.5 · 2/-1.5 = 10
b = 0-c-d+f+x/2 = 0-10-12+36+2/2 = 8
a = 0+b-x = 0+8-2 = 6

a = 6
b = 8
c = 10
d = 12
f = 36
x = 2

Our linear equations calculator calculates it.

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