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 $x = \frac {C- B} A$.

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 $A = A_1 - A_2$.

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 watch the following YouTube that explores this connection for the equation $5x-7=8$ with much detail.

Here is another example demonstrating that connection:

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 $g$).

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 $y$.