Understanding Consistent Linear Systems and Identifying Infinitely Many Solutions

a consistent linear system must have infinitely many solutions

A consistent linear system is a system of linear equations that has at least one solution, meaning there is at least one set of values for the variables that satisfy all the equations simultaneously

A consistent linear system is a system of linear equations that has at least one solution, meaning there is at least one set of values for the variables that satisfy all the equations simultaneously.

If a consistent linear system has infinitely many solutions, it means that there are infinitely many sets of values for the variables that will satisfy the equations. This occurs when the equations are dependent on each other, meaning one or more equations can be obtained by adding or subtracting multiples of the other equations. As a result, the system has more variables than necessary to uniquely determine a solution, allowing for an infinite number of possible solutions.

One way to determine if a consistent linear system has infinitely many solutions is by using the concept of rank. The rank of a system represents the maximum number of linearly independent rows or columns in the system’s augmented matrix. If the rank is less than the number of variables in the system, then there must be infinitely many solutions.

Another way to identify a consistent linear system with infinitely many solutions is by observing that the equations are proportional or equivalent. This means that any linear combination of solutions to the system will also be a solution. In other words, if (x1, y1, z1) is a solution, then (kx1, ky1, kz1) will also be a solution for any non-zero value of k.

Overall, a consistent linear system must have at least one solution, but if it has infinitely many solutions, it implies that the equations are not sufficient to uniquely determine the values of the variables.

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