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7980Proposition: Presentation of a Straight Line in a Plane as a Linear Equation

Let $L$ be a straight line in the number plane $\mathbb R^2.$ Then

  • $L$ goes through the origin $(0,0)\in L$ if and only if $L$ is the set of solutions of the homogenous linear equation $\alpha_1x_1+\alpha_2x_2=0$ for some real-valued coefficients $\alpha_1,\alpha_2\in\mathbb R$ which do not both equal zero.
  • $L$ does not go through the origin $(0,0)\not\in L$ if and only if $L$ is the set of solutions of the inhomogenous linear equation $\alpha_1x_1+\alpha_2x_2=\beta$ for some real-valued coefficients $\alpha_1,\alpha_2\in\mathbb R$ which do not both equal zero and some real number $\beta\in\mathbb R$, $\beta\neq 0.$

| | | | | Contributors: bookofproofs | References: [7937]

79811.Proof: (related to "Presentation of a Straight Line in a Plane as a Linear Equation")


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Bibliography (further reading)

[7937] Knabner, P; Barth, W.: “Lineare Algebra – Grundlagen und Anwendungen”, Springer Spektrum, 2013

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