Simultaneous Equation Puzzles

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Puzzle for January 4, 2019  ( )

Scratchpad

Find the 6-digit number ABCDEF by solving the following equations:

eq.1) A + B + C + D + E + F = 21 eq.2) A = C + E eq.3) A + E = F eq.4) E - C = B - F eq.5)* A + B + BC + F = DE eq.6)* EF - D = B + D + F

A, B, C, D, E, and F each represent a one-digit non-negative integer.
*  BC, DE, and EF are 2-digit numbers (not B×C, D×E, or E×F).

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Hint #1

In eq.3, replace A with C + E (from eq.2): C + E + E = F which becomes eq.3a) C + 2×E = F In eq.4, substitute (C + 2×E) for F: E - C = B - (C + 2×E) which is the same as E - C = B - C - 2×E Add C + 2×E to both sides of the equation above: E - C + C + 2×E = B - C - 2×E + C + 2×E which makes 3×E = B

  

Hint #2

eq.6 may be written as: 10×E + F - D = B + D + F Add (D - B - F) to both sides of the equation above: 10×E + F - D + (D - B - F) = B + D + F + (D - B - F) which simplifies to 10×E - B = 2×D Substitute 3×E for B: 10×E - 3×E = 2×D which makes 7×E = 2×D Divide both sides by 2: 7×E ÷ 2 = 2×D ÷ 2 which makes 3½×E = D

  

Hint #3

eq.5 may be written as: A + B + 10×B + C + F = 10×D + E which is equivalent to A + 11×B + C + F = 10×D + E Substitute (3×E) for B, and (3½×E) for D in the above equation: A + 11×(3×E) + C + F = 10×(3½×E) + E which becomes A + 33×E + C + F = 35×E + E Subtract 33×E from each side: A + 33×E + C + F - 33×E = 35×E + E - 33×E which becomes eq.5a) A + C + F = 3×E

  

Hint #4

In eq.5a, substitute C + 2×E for F (from eq.3a): A + C + C + 2×E = 3×E Subtract 2×E from each side of the above equation: A + C + C + 2×E - 2×E = 3×E - 2×E which becomes A + 2×C = E In eq.2, substitute A + 2×C for E: A = C + A + 2×C Subtract A from both sides: A - A = C + A + 2×C - A which simplifies to 0 = 3×C which means 0 = C

  

Hint #5

Substitute 0 for C in eq.2: A = 0 + E which means A = E

  

Hint #6

In eq.3, replace A with E: E + E = F which makes 2×E = F

  

Solution

Substitute E for A, 3×E for B, 0 for C, 3½×E for D, and 2×E for F in eq.1: E + 3×E + 0 + 3½×E + E + 2×E = 21 which simplifies to 10½×E = 21 Divide both sides of the equation above by 10½: 10½×E ÷ 10½ = 21 ÷ 10½ which makes E = 2 making A = E = 2 B = 3×E = 3 × 2 = 6 D = 3½×E = 3½ × 2 = 7 F = 2×E = 2 × 2 = 4 and ABCDEF = 260724