Chapter 6, Problem 5MCQ

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 5MCQ

The given system of equations has infinitely many solutions which are dependent.

Explanation of Solution

Given information:

The given system of equations is

  $x+y=8$

  $3x+3y=24$

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 graphs of given equations $x+y=8$ , and $3x+3y=24$ are shown in Figure-1 here.

From the graph of equations we see that the equation $x+y=8$ and $3x+3y=24$ represent same straight lines so they have infinitely many solutions, and those solutions are dependent.

Thus, the given system of equations $x+y=8$ , $3x+3y=24$ has infinitely many dependent solutions.

We discuss here what the infinitely many solutions are:

See equation $\frac{3x+3y}{3}=\frac{24}{3}\Rightarrow x+y=8$ that is the first equation. Now, let $x=k$ , k be a real number. Substituting $x=k$ in first equation $x+y=8$ we get $y=8-k$ ,

Thus, the solutions of given system of equations is $x=k$ and $y=8-k$ . Since, we have infinite values of k so we get an infinite number of solutions. Since, both x and y depends on k so the solutions are dependent.

Conclusion:

The given system of equations $x+y=8$ , $3x+3y=24$ has infinitely many dependent solutions.