Question

sketch the graph or use a graphing utility to determine which of the following solution scenarios is true for the system of equations.

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(1 point)
two of the planes are parallel, which means there are no solutions.
all three planes intersect at a single point, which means there is one solution.
all three planes are parallel, which means there are no solutions.
all three planes overlap, which means there are infinite solutions.

Explanation:

Step1: Write the equations in matrix form

The system of equations can be written as \(A\vec{x}=\vec{b}\), where \(A=

$$\begin{bmatrix}7&3&- 1\\14&6&-2\\-7&-3&1\end{bmatrix}$$

\), \(\vec{x}=

$$\begin{bmatrix}x\\y\\z\end{bmatrix}$$

\) and \(\vec{b}=

$$\begin{bmatrix}12\\20\\0\end{bmatrix}$$

\)

Step2: Check the relationship between the rows of the coefficient matrix

Multiply the first row of \(A\) by \(2\): \(2\times(7,3, - 1)=(14,6,-2)\), which is the second row of \(A\).
Add the first and the third row: \((7 + (-7),3+(-3),-1 + 1)=(0,0,0)\)

Step3: Check the relationship between the augmented matrix \([A|\vec{b}]\)

Multiply the first - row of \([A|\vec{b}]\) by \(2\): \(2\times(7,3,-1,12)=(14,6,-2,24)\)
The second - row of \([A|\vec{b}]\) is \((14,6,-2,20)\). Since \(24
eq20\), the system is inconsistent.

Answer:

Two of the planes are parallel, which means there are no solutions.