This is not the only concept of "linear" - see further down the page.
A straight line in two dimensional geometry has the characteristic that whenever you increase (or decrease) the variable $x$ by a particular amount, the variable $y$ will change by some multiple of that amount, no matter what the values actually are. So no matter what the values of $x$ and $y$ we have "change in y" is a constant times "change in x."

If $y$ is 0 when $x$ is 0, that means we have:

$y~=~kx$

In this case, $k$ is the ratio of changes.

In general, when $x=0$ the vaue of $y$ will not necessarily be $0.$ As a result, a more general form is:

$y~=~kx+c$

If there is a point, $(x_0,y_0)$ that the line goes through, that means that when $x=x_0$ we must have $y=y_0.$ As a result, we have this form:

$(y-y_0)~=~k(x-x_0)$

Using algebra to rearrange that equation, we have:

$y=kx+(y_0-kx_0)$

This is now in the same form as $y~=~kx+c.$

The constant $k$ in these equations is the "slope" of the line, and often the letter $m$ is used to represent the slope, rather than $k.$

Other definitions of "Linear"

In the sense of "Linear Algebra" a function $f$ is said to be "linear" if it satisfies the following:

A consequence of these conditions is that $f(0)$ must be $0.$ To see this, consider the first condition with $x=k$ and $y=-k.$

Linear functions don't need to be just from $\mathbb{R}$ to $\mathbb{R}$ or $\mathbb{C}$ to $\mathbb{C}$ but can be from any set to any set, provided it has suitable concepts of addition and multiplication - such a set forms a vector space. In this way we can then start to represent linear functions by matrices. This is why "Linear Algebra" is really the study of matrices, and matrix multiplication.

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