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How do I teach the difference between Linear Equations and Equation of a Line

Etymology/Grammar "Line" is the root word, a noun whereas "linear" is a derived word, an adjective. So, apart from the grammar distinction, one could call "line" and &...
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How do I teach the difference between Linear Equations and Equation of a Line

I suspect that two issues may be partly underlying your intuition: First, the equation $y = mx + c$ gives $y$ as an explicit function of $x$, whereas $ax + by + c = 0$ only gives a relation between $...
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How do I teach the difference between Linear Equations and Equation of a Line

Your intuition is fundamentally not correct. These are all equations of a line, and your textbook is correct to include them in the same chapter. The term "Linear Equation" is used because ...
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How do I teach the difference between Linear Equations and Equation of a Line

They aren't very much different from some perspectives. If what you care about is the question "Is this point on the line or not?" then you can simply plug your $(x,y)$ values into either of ...
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How do I teach the difference between Linear Equations and Equation of a Line

The only "real" difference (except for maybe some analogies that work differently as in your house/street example) is that the linear equations are a proper superset of the equations of ...
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