Consider the linear function $f(x) = mx + b$.

A standard form of a linear equation with variable $x$ is an equation of the form $Ax + B = C$ with $A \ne 0$.
Beginning algebra spends a considerable amount of time solving this equation without any reference to functions.
When $B = 0$ this equation has the solution $x = \frac C A$.
When $B \ne 0$ the equation can be solved so that $x = \frac {C- B} A$.

Example  LEQ.1 : Suppose $5x - 10 = 20$. Find $x$.
A slightly more ambitious form of a linear equation with variable $x$ is an equation of the form $A_1x + B = A_2x +C$ with $A_1 \ne A_2$. Algebraically this is solved by solving the related equation $Ax + B = C$ where $A = A_1 - A_2$.

Example LEQ.2 : Suppose $5x - 10 = 3x + 20$. Find $x$..
There is an important connection between the algebraic steps for solving a linear equation and a mapping diagram for a linear function as a composition.
You can watch the following YouTube that explores this connection for the equation $5x-7=8$ with much detail.

Here is another example demonstrating that connection:
Example LEQ.3 : Suppose $2x + 1 = 2$. Find $x$.

You can use this next dynamic example to solve linear equations like those in Examples LEQ.1 and LEQ.2 visually with focus points on a mapping diagram of $f$ (and $g$) and the lines in the graph of $f$ (and $g$).

Example LF.DLEQ.0 Dynamic Views for solving an equation $f(x) = mx+b = c$ on Graphs and Mapping Diagrams

Example LF.DLEQ2.0 Dynamic Views for solving equality of two linear functions, $f(x) = g(x)$, where $f(x)= m_fx+b_f$ and $g(x)= m_gx+b_g$ on Graphs and Mapping Diagrams

LINEAR Equations with 2 Variables

A linear equation with variables $x$ and $y$ is an equation of the form $Ax + By = C$ with $A^2 + B^2 \ne 0$.
When $B = 0$ this equation has the solution $x = - \frac C A$.
When $B \ne 0$ the equation can be solved so that $y= f(x) = - \frac A B x + \frac C B$. So $y$ is a linear function, $f(x)$, where $m = -\frac A B$ and $b = \frac C B$.

Questions involving linear equations usually involve a second piece of information (another linear equation)  to have a unique solution.

Example LF.LEQ.4. Suppose $5x - 10y = 20$ (i) and $x=8$ (ii). Find $y$.

Example LF.LEQ.5. Suppose $5x - 10y = 20$ (i) and $y=8$ (ii). Find $x$.

Example LF.LEQ.6. Suppose $5x - 10y = 20$ (i) and $x+y=1$ (ii). Find $x$ and $y$.