Use cramer's rule to solve -0.4x_1+0.8x_2=1.6 0.2x_1+0.3x_2=0.6

Anish Buchanan 2021-01-13 Answered
Use cramers
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Answered 2021-01-14 Author has 88 answers

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I have a representation of a linear equation in standard form (ax+by+c=0) which I am representing as a set of coefficients: a,b,c.
I want to normalize these so that any two equations that represent the same line can be compared programatically to see if they're equal just using the co-efficients. For this I need the co-efficients of any two equations representing the same line to be equivalent after normalizing.
I normalized the values by dividing all coefficients by a2+b2+c2 which scales the coefficients to be the same but I don't know how to account for sign changes in this.
For example, if I have equations: 2x+-4y+2=0 (represented as 2, -4, 2) and -2x+4y+-2=0 (represented as -2, 4, -2), how can I transform the coefficients to make them equal?
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IMHO, the alternative of going through solving systems of linear equations beforehand, obscures the topic.
What would be equivalent, slightly abstract approaches to study analysis for a freshman? A bit of topology and metric spaces, like Rudin? Multivariate calculus with differential forms, like Hubbard & Hubbard?
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Consider the equation x+3=3(x+1)2x. Is this equation a linear equation?
I would say to my students that an equation in 1 variable is linear when it can be simplified to the form ax=b where a0. Also, I would mention that a linear equation in 1 variable has only 1 solution. Therefore, I would say that the equation x+3=3(x+1)2x is not linear because it can be reduced to 0x=0 which has infinite solutions.
However, a colleague whose research area is algebra says that a linear equation is one that only involves polynomials of degree 1. Therefore, the equation x+3=3(x+1)2x is linear. Then, according to the definition of my colleague, the equation xx=0 is linear as well.
I was thinking that maybe a third possibility is that according to the definition of "linear equation" only equations of the form "expression=0" can be classified as linear or non-linear. In this case, the equation x+3=3(x+1)2x is not linear nor non-linear, but the equation xx=0 would be linear.
What should be the definition for "linear equation" in 1 variable suitable for a math course?