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Q3Q

Expert-verifiedFound in: Page 1

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.

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.

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 \(3 \times 3\).

**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.

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