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### Linear Algebra and its Applications

Book edition 5th
Author(s) David C. Lay, Steven R. Lay and Judi J. McDonald
Pages 483 pages
ISBN 978-03219822384

# 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’)

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

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

(c) The inconsistent linear system of three variables and equations has no common point.

See the 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,

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 of three 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 $$3 \times 3$$.

## (c) Step 3: Write the condition when the graphs have no point in common

If there is no common point between the planes, their intersection is not a unique line or point. As a result, there is no way to solve it.

Thus, the inconsistent linear system of three variables and equations has no common point.