Linear Equations

An equation is linear if all variables present have powers of 1 (no squares, no cubes, etc.)

A linear equation with one variable fits the form ax+b=0.

We can write a general equation using a’s for constants and x’s for variables:

$a_1x_1 + a_2x_2 + ... + a_{n-1}x_{n-1} + a_nx_n + b = 0$

Every variable has a power of 1; there are no squares or higher powers. The formula below, where symbols {a,b,c,d} are constants, could be used as a linear equation in 3D space:

$ax + by + cx + d = 0$

A line has a slope in every dimension.

Appendix A

We know that the graph of y=mx + b will result in a line.

What about ax+by=c?

We will try to get the above in a form like “y=mx+b” using algebraic manipulation.

$\dfrac{a}{b}x+\dfrac{b}{b}y=\dfrac{c}{b}$

$\dfrac{a}{b}x+y=\dfrac{c}{b}$

$y= -\dfrac{a}{b}x + \dfrac{c}{b}$

Hopefully you can see that we have it, with m = -(a/b) and b = (c/b).

Appendix A

If you are interested in our “Dark Blue Magic” program, try thinking of “ax + b = 0” as being the following:

$ax^1 + bx^0 = 0$

Or even better, rewrite the constants, a & b:

$q_1 = a$
$q_2 = b$

We now have

$q_1x^1 + q_0x^0 = 0$

We can write this as a Summation:

$\sum_{n=0}^{1} q_i \: x_i=0$

Appendix B

Some caution is needed, as there are three different topics that make use of the word linear:

Linear Equations
Linear
System of Linear Equations

Appendix C

The equation

w = ax + by + cz

is a line in 3D space. When we add variables, we add dimensions to the space.

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