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What is the most general solution?
33x + 55y + 79z - 99w = r
The solution requires x , y , z , and w
to be expressed in terms of r , s , t , u .
The general solution of the indeterminate equation
will be in the form :
x = A₁r + B₁s + C₁t + D₁u
y = A₂r + B₂s + C₂t + D₂u
z = A₃r + B₃s + C₃t + D₃u
w= A₄r + B₄s +C₄t+D₄u
2 Answers
- knashhaLv 51 decade agoFavorite Answer
One simple way is reduce mod 11:
79z = r = 2z mod11, z = 6r mod 11 so set z = 11s + 6r
Put this value into the equation and divide by 11 to get,
3x + 5y - 9w = -43r-79s. (this is the "new" r value). Let's reduce mod3 to get
5y = -r -s mod3, 2y = -r-s mod3, so y = r + s mod 3 and we
can write,
y = 3t + r + s. Plug this in and divide by 3 to get.
x - 3w = -5t - 16 r - 28s and reduce mod 3
x = t - r - s mod3 so we introduce a new variable u and,
x = 3u + t - r - s. Plug in and divide by 3 and solve for w
w = u + 2t + 9s + r, together with the values,
x = 3u + t - r - s
y = 3t + r + s
z = 11s + 6r
provide a parametric solution to your equation.
