Answered: For two linear equations in three…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

For two linear equations in three unknowns x, y, z, the row picture will show (2 or 3) (lines or planes) in (2 or 3)-dimensional space. The column picture is in (2 or 3)dimensional space. The solutions normally lie on a __ .

Expert Solution

Step 1

Given that there are two linear equations in three unknowns x,y,z.

Suppose, the two linear equations are:

$\begin{array}{rcl} a_{1} x + b_{1} y + c_{1} z & = & d_{1} \\ a_{2} x + b_{2} y + c_{2} z & = & d_{2} \end{array}$

If row picture is considered, then the matrix is:

$[\begin{matrix} a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \end{matrix}] \times [\begin{matrix} x \\ y \\ z \end{matrix}] = [\begin{matrix} d_{1} \\ d_{2} \end{matrix}]$

So, there will be $2$ planes in three-dimensional space.

The intersection of the planes will be a line.

So, the solutions lie on a LINE.

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