Chapter 1: Q3Q (page 1)

The solutions \(\left( {x,y,z} \right)\) of a single linear equation \(ax + by + cz = d\)
form a plane in \({\mathbb{R}^3}\) when a, b, and c are not all zero. Construct sets of three linear equations whose graphs (a) intersect in a single line, (b) intersect in a single point, and (c) have no points in common. Typical graphs are illustrated in the figure.
Three planes intersecting in a line.
(a)
Three planes intersecting in a point.
(b)
Three planes with no intersection.
(c)
Three planes with no intersection.
(c’)

Short Answer

Expert verified

(a) The echelon form of the consistent linear system is,,or.

(b) The echelon form of the consistent linear system isthe identity matrix of the order\(3 \times 3\).

Step by step solution

(a) Step 1: Write the condition when the graphs intersect on a single line

Each point on the line is a solution to the given system of equations. And the solution set is infinite if the three planes cross at a single point. As a result, there must be two pivot components in the possible forms.

The echelon form of the consistent linear system is shown below:

Or,

Here, is the leading entry, and \(\left( * \right)\) can have any value, including 0.

(b) Step 2: Write the condition w\(3 \times 3\)hen the graphs intersect at a single point

The system ofthree equations is fulfilled if the three planes cross at a single location. As a consequence, the system is consistent, and it offers a unique solution.

The echelon form that may be produced by solving this system of equations is an identity matrix of the order .

Thus, the echelon form of the consistent linear system is an identity matrix of the order \(3 \times 3\).