Q35E Consider two distinct points聽 [... [FREE SOLUTION]

91影视

Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 6: Q35E (page 290)

Consider two distinct points $[\begin{matrix} a_{1} \\ a_{2} \end{matrix}]$ androle="math" localid="1660316333459" $[\begin{matrix} b_{1} \\ b_{2} \end{matrix}]$ in the plane. Explain why the solutions $[\begin{matrix} x_{1} \\ x_{2} \end{matrix}]$ of the equation $d e t [\begin{matrix} 1 & 1 & 1 \\ x_{1} & a_{1} & b_{1} \\ x_{2} & a_{2} & b_{2} \end{matrix}] = 0$ form a line and why this line goes through the two pointsrole="math" localid="1660316534303" $[\begin{matrix} a_{1} \\ a_{2} \end{matrix}]$ and $[\begin{matrix} b_{1} \\ b_{2} \end{matrix}]$ .

Short Answer

Expert verified

Therefore, $[\begin{matrix} x_{1} \\ x_{2} \end{matrix}] = [\begin{matrix} a_{1} \\ a_{2} \end{matrix}]$ and $[\begin{matrix} x_{1} \\ x_{2} \end{matrix}] = [\begin{matrix} b_{1} \\ b_{2} \end{matrix}]$ are a solution.

Step by step solution

Step by Step Solution: Step 1: Given

Consider two distinct points are,

$[\begin{matrix} a_{1} \\ a_{2} \end{matrix}]$

$[\begin{matrix} b_{1} \\ b_{2} \end{matrix}]$

To solve equations

We solve,

$d e t [\begin{matrix} 1 & 1 & 1 \\ x_{1} & a_{1} & b_{1} \\ x_{2} & a_{2} & b_{2} \end{matrix}] = 0$

$a_{1} b_{2} + b_{1} x_{2} + a_{2} x_{1} - a_{1} x_{2} - a_{2} b_{1} - b_{2} x_{1} = 0$

$(a_{2} - b_{2}) x_{1} + (- a_{1} + b_{1}) x_{2} + (a_{1} b_{2} - a_{2} b_{1}) = 0$

This is a linear equation. By computing, we can see that:

role="math" localid="1660317162856" $[\begin{matrix} x_{1} \\ x_{2} \end{matrix}] = [\begin{matrix} a_{1} \\ a_{2} \end{matrix}]$

And

role="math" localid="1660317188227" $[\begin{matrix} x_{1} \\ x_{2} \end{matrix}] = [\begin{matrix} b_{1} \\ b_{2} \end{matrix}]$ Satisfy the equation.

Therefore, $[\begin{matrix} x_{1} \\ x_{2} \end{matrix}] = [\begin{matrix} a_{1} \\ a_{2} \end{matrix}]$ androle="math" localid="1660317351692" $[\begin{matrix} x_{1} \\ x_{2} \end{matrix}] = [\begin{matrix} b_{1} \\ b_{2} \end{matrix}]$ are a solution.

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Consider two distinct points [a1a2]androle="math" localid="1660316333459" [b1b2] in the plane. Explain why the solutions [x1x2] of the equation det[111x1a1b1x2a2b2]=0 form a line and why this line goes through the two pointsrole="math" localid="1660316534303" [a1a2] and [b1b2].

Short Answer

Step by step solution

Step by Step Solution: Step 1: Given

To solve equations

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

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Applied Mathematics

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