A system of equations may have a unique solution, an infinite number of solutions, or no solution. Use matrices to find the general solution of the following system, if a solution exists.
A) , ,
B) , ,
C) , ,
D) , , for any real number of z.
E) no solution

Question

A system of equations may have a unique solution, an infinite number of solutions, or no solution. Use matrices to find the general solution of the following system, if a solution exists. 
A) , , 
B) , , 
C) , , 
D) , , for any real number of z. 
E) no solution

Quizplus · Accepted Answer

We can write the system of equations as an augmented matrix:

$$\left[\begin{array}{cc|c} 1 & -2 & 3 \ 2 & -4 & 6 \ 3 & -6 & 9  \end{array}ight]$$

We can perform row operations to transform this matrix into reduced row echelon form:

$$\left[\begin{array}{cc|c} 1 & -2 & 3 \ 0 & 0 & 0 \ 0 & 0 & 0  \end{array}ight]$$

This corresponds to the system:

\begin{align*}
x - 2y &= 3 \
0 &= 0 \
0 &= 0
\end{align*}

The first equation gives us $x = 2y + 3$. So the general solution is:

$$\begin{pmatrix} x \ y \ z \end{pmatrix} = \begin{pmatrix} 2y+3 \ y \ z \end{pmatrix} = \begin{pmatrix} 2 \ 1 \ 0 \end{pmatrix}y + \begin{pmatrix} 3 \ 0 \ 1 \end{pmatrix}z$$

So the system has infinitely many solutions, and the correct answer is D.

A System of Equations May Have a Unique Solution, an Infinite