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: ExampleLEQ.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 $g$).
ExampleLF.DLEQ.0Dynamic
Views for solving an equation $f(x) = mx+b = c$ on Graphs and
Mapping Diagrams
ExampleLF.DLEQ2.0Dynamic
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 $y$.