The black and white signal Y is defined as
0.299 * Red + 0.587 * Green + 0.114 * Blue.
Or with more rounded numbers: Y = 0.30 * R + 0.59 * G + 0.11 * B.
Let's look at;
0.299 * (R-Y) + 0.587 * (G-Y) + 0.114 * (B-Y) = ?
Or, with more rounded numbers;
0.30 * (R-Y) + 0.59 * (G-Y) + 0.11 * (B-Y) = ?
This is the same as;
0.30*R - 0.30*Y + 0.59*G - 0.59*Y + 0.11*B - 0.11*Y
Grouping the Y-bits together;
0.30*R + 0.59*G + 0.11*B - (0.30*Y + 0.59*Y + 0.11*Y)
And since 0.30 + 0.59 + 0.11 = 1;
0.30*R + 0.59*G + 0.11*B - Y
And Y is defined as 0.30*R + 0.59*G + 0.11*B. So it actually says 'Y - Y'!
Which of course is zero.
And therefore;
0.30 * (R-Y) + 0.59 * (G-Y) + 0.11 * (B-Y) = 0
If we subtract '0.30 * (R-Y)' and '0.11 * (B-Y)' at both sides of the equal sign we get;
0.59 * (G-Y) = -0.30 * (R-Y) -0.11 * (B-Y)
Next divide both sides by 0.59;
0.59 0.30 0.11 ────── * (G-Y) = - ────── * (R-Y) - ────── * (B-Y) 0.59 0.59 0.59
Which yields;
G-Y = - 0.51 * (R-Y) - 0.19 * (B-Y)