Chapter 6, Problem 2PT

To determine

To find: The number of solutions of given system of equations using graph and if it has solution, name it.

Answer to Problem 2PT

The given system of equations has only one solution $x=1$ , $y=4$ , the solution is independent.

Explanation of Solution

Given information:

The given system of equations is

  $y=x-3$

  $y=-2x+9$

Formula used:

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.

Graph:

The graph of given equations $y=x-3$ , and $y=-2x+9$ is shown in Figure-1 here.

From the graph of equations we see that the lines intersect at a point (1, 4), therefore, $x=1$ , $y=4$ is the solution of equations $y=x-3$ and $y=-2x+9$ .

Thus, the given system of equations has only one solution $x=1$ , $y=4$ so is consistent, and as there is only one solution so the solution is independent.

Conclusion:

The given system of equations has only one solution $x=1$ , $y=4$ , the only solution is independent.