Chapter 6, Problem 10SGR

To determine

To answer: Does a consistent and independent system of equations have an infinite number of solutions.

Answer to Problem 10SGR

A consistent and independent system of equations has finite number of solutions (one).

Explanation of Solution

Given information:

The given system of equations is consistent and independent.

If a system has at least one solution, then it is said to be consistent. If a consistent system of equations has exactly one solution, it is independent .

If a consistent system has an infinite number of solutions, it is dependent, when graphed such equations, represent the same line. If a system has no solution, it is said to be inconsistent .The graphs of such equations do not intersect, i.e., graphs are parallel. Consider system of equations $x+4y=6$ , and $\frac{1}{2}x+2y=3$ is shown in Figure-1 here.

The graph of both the equations is same, the solution of equation $x+4y=6$ is also solution of equation $\frac{1}{2}x+2y=3$ i.e. the solutions are dependent. There are infinite number of solutions. Since system has solution so system is consistent.

Thus, a consistent and dependent system of equations has infinite number of solutions, and a consistent and independent system of equations has finite number of solutions (one).

Conclusion:

A consistent and independent system of equations has finite number of solutions (one).