How can we make sure that the line formed on a Cartesian plane by a linear equation is always straight? I mean, that whenever we talk about linear equations, we say that they always form a straight line on a graph.  We make this sure by making a graph of some possible solutions to the equation and plotting the points on the graph.  But can we make sure that all the solutions of the linear equation will always form a straight line because we cannot plot each and every solution of the equation on the graph because there are infinitely many solutions to a linear equation in $2$ variables? 
 A: As you shall see you have asked a deep question.
A linear equation
$$ax+by=c\tag{1}$$ with $(a,b)$ assumed $\ne(0,0)$ defines a solution set
$$L:=\bigl\{(x,y)\in{\mathbb R}^2\bigm| ax+by+c\bigr\}\ .\tag{2}$$
If, say, $b\ne0$ then we can solve $(1)$ for $y$ and obtain $y={1\over b}(c-ax)$, whereby $x\in{\mathbb R}$ can be arbitrary. In this way the implicit description $(2)$ of $L$ can be replaced by the explicit presentation
$$L=\left\{{\bf z}\in{\mathbb R}^2\biggm|{\bf z}=\left(x,{1\over b}(c-ax)\right), \  x\in{\mathbb R}\right\}\ ,$$
which produces for each $x\in{\mathbb R}$ a point ${\bf z}\in L$ in a bijective way.
Where are the "straight lines" here? The simple answer is: You have not defined what you mean by a "straight line". Some people might say: A straight line is a set $L\subset{\mathbb R}^2$ defined by an equation of type $(2)$. But maybe you are more at home in "elementary Euclidean geometry". In this case there is a long way from "straight lines" to linear equations.
A: One way to see this is to consider the slope of a line. This is a measure of how steep the line is. It can be defined by taking two points on the line $(x_1,y_1)$ and $(x_2,y_2)$ and calculating $$m=\frac{y_2-y_1}{x_2-x_1}.$$ You should convince yourself that no matter which two points you pick on a straight line, the proportion of vertical change to horizontal change is constant! Now let $(a,b)$ be a fixed point on a line with slope $m$. Let $(x,y)$ be a point that varies. Plugging into our formula for slope, we have:
$$m=\frac{y-b}{x-a}.$$ Multiplying by $(x-a)$, we get $m(x-a)=y-b,$ which is a linear equation in $x,y$. Moreover, any linear equation (except $x=k$) arises in this way. 
So, to recap, start with a straight line. It has constant slope. Write down the equation for constant slope, and you get a linear equation. Any linear equation (except a vertical one, which has undefined slope) can be gotten in this way.
