Independent equation
![The equations x − 2y = −1, 3x + 5y = 8, and 4x + 3y = 7 are linearly dependent, because 1 times the first equation plus 1 times the second equation reproduces the third equation. But any two of them are independent of each other, since any constant times one of them fails to reproduce the other.](/Images/godic/202501/20/Three_Intersecting_Lines.svg1603.png")
![The equations 3x + 2y = 6 and 3x + 2y = 12 are independent, because any constant times one of them fails to produce the other one.](/Images/godic/202501/20/Parallel_Lines.svg1603.png")
![A system of three linearly independent equations, y=x+1, y=–2x+1, and y=3x–2. There are no two constants a and b such that a times the first equation plus b times the second equation equals the third equation.](/Images/godic/202501/20/3_equations_-11603.JPG")
An independent equation is an equation in a system of simultaneous equations which cannot be derived algebraically from the other equations. The concept typically arises in the context of linear equations. If it is possible to duplicate one of the equations in a system by multiplying each of the other equations by some number (potentially a different number for each equation) and summing the resulting equations, then that equation is dependent on the others. But if this is not possible, then that equation is independent of the others.