How do you tell if a system of equations has a unique solution?

A nxn homogeneous system of linear equations has a unique solution (the trivial solution) if and only if its determinant is non-zero. If this determinant is zero, then the system has an infinite number of solutions.

Can a system of equations with a free variable have a unique solution?

No there should not be a system of three equations with 4 unknowns with a unique solution. In order to have a unique solution you need at least as many equations equations as variables.

Can a linear system with more unknowns than equations have a unique solution?

In general, a system with the same number of equations and unknowns has a single unique solution. In general, a system with more equations than unknowns has no solution.

How many consistent independent equations are needed to solve for unknown variables?

With only two equations and three independent unknowns, this can be solved. You can’t prove this for the general case, because it is possible to have a set of n-1 equations in n variables where the set of possible solutions is a single point in an n-dimensional coordinate space.

Which systems of equations have infinite solutions?

An independent system of equations has exactly one solution (x,y) . An inconsistent system has no solution, and a dependent system has an infinite number of solutions.

Is any ordered pair in a system that makes all the equations of that system true?

In general, a solution of a system in two variables is an ordered pair that makes BOTH equations true. In other words, it is where the two graphs intersect, what they have in common. So if an ordered pair is a solution to one equation, but not the other, then it is NOT a solution to the system.

Does the system have a unique solution no solution or many solutions?

The three types of solution sets: A system of linear equations can have no solution, a unique solution or infinitely many solutions. A system has no solution if the equations are inconsistent, they are contradictory. for example 2x+3y=10, 2x+3y=12 has no solution. is the rref form of the matrix for this system.

Can a homogeneous system have a unique solution?

This is called the Trivial Solution. Since a homogeneous system always has a solution (the trivial solution), it can never be inconsistent. Thus a homogeneous system of equations always either has a unique solution or an infinite number of solutions.

Why does a system need to have at least as many equations as unknowns to have a unique solution?

Some linear systems may not have a solution, while others may have an infinite number of solutions. In order for a linear system to have a unique solution, there must be at least as many equations as there are variables. since it makes all three equations valid.

Is the system independent dependent or inconsistent?

If a consistent system has exactly one solution, it is independent . If a consistent system has an infinite number of solutions, it is dependent . If a system has no solution, it is said to be inconsistent . The graphs of the lines do not intersect, so the graphs are parallel and there is no solution.

Which systems of equations have no solution?

What equation has no solution?

A system of linear equations has no solution when the graphs are parallel. Infinite solutions. A system of linear equations has infinite solutions when the graphs are the exact same line.