What happens when you have more equations than unknowns?

In mathematics, a system of equations is considered overdetermined if there are more equations than unknowns. An overdetermined system is almost always inconsistent (it has no solution) when constructed with random coefficients. Such systems usually have an infinite number of solutions.

Can a system with more equations than unknowns be consistent?

A system is consistent when there is no “contradictory statements.” But if we have more equations than variables, then the system itself can be either redundant or inconsistent.

Can a system with more variables than equations have a unique solution?

If there are more variables than equations, you cannot find a unique solution, because there isnt one. However, you can eliminate some of the variables in terms of others. In other words, you can start the Gaussian elimination process and continue until you run out of rows.

How many solutions does a homogeneous system of linear equations with more unknowns than equations has?

Every homogeneous system has either exactly one solution or infinitely many solutions. If a homogeneous system has more unknowns than equations, then it has infinitely many solutions.

Can a linear system with fewer equations than unknowns have no solution explain?

A system of linear equations with fewer equations than unknowns is sometimes called an underdetermined system. Can such a system have a unique​ solution? Explain. ​No, it cannot have a unique solution.

How many solutions does a consistent system of linear equations have if there are more unknowns than equations?

A system has infinitely many solutions when it is consistent and the number of variables is more than the number of nonzero rows in the rref of the matrix. For example if the rref is has solution set (4-3z, 5+2z, z) where z can be any real number.

Can a system with fewer equations than variables have a solution?

A system of m × n linear homogeneous equations with fewer equations than unknowns (m < n) has at least one free variable, hence an infinite number of solutions. Therefore, such a system always has the zero solution and also a nonzero solution.

Why a system of linear equations with more unknowns than equations has infinite many solution?

(b) A homogeneous system of 5 equations in 4 unknowns. Since the system is homogeneous, it has the zero solution. Since there are more equations than unknowns, we cannot determine further. Thus the possibilities are either a unique solution or infinitely many solution.

Why a system of linear equations with more unknowns then equations has infinite many solution?

A system of m×n linear homogeneous equations with one unknown missing has at least one free variable, hence an infinite number of solutions.