Chapter 6, Problem 12SGR

To determine

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

Answer to Problem 12SGR

The given system of equations has one solution $x=1$ , $y=-2$ , and the system is independent.

Explanation of Solution

Given information:

The given system of equations is

  $y=2x-4$

  $4x+y=2$ .

Concept used:

The point of intersection of two lines is the solution of equations.

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 $y=2x-4$ , and $4x+y=2$ is shown in Figure-1 here.

From the graph of equations we see that the lines $y=2x-4$ and $4x+y=2$ intersect at point $\left(1,-2\right)$ i.e. the given system of equations has a solution $x=1$ , $y=-2$ , so is consistent. Since, the given system of equations has only one solution so the system is independent.

Conclusion:

The given system of equations has one solution $x=1$ , $y=-2$ , and the system is independent.