Linear function are at the heart of many mathematical applications: from simple scales, rates, and proportions to estimating solutions of equations and differential equations.

The conventional representation of a linear function as $f(x) = mx + b$ can sometimes be missing in applications, but that omission cannot diminish its importance. Visualizing these applications with cartesian graphs sometimes is also missed. In this section we highlight a few of the more elementary applications and their visualization with mapping diagrams.

Topics: [ Use knowls]
Speed (Velocity)
Direct Proportion

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  LAPP.1 : Motion and Speed (Velocity) $s(t)= vt + s_0$ where $s$ is position of an object moving on a path (often a straight line), $v$ is the (constant) speed (velocity) of the moving object, $t$ is the elapsed time that the object has been in motion, and $s_0$ is the position of the object at time $t=0$.

Example LAPP.2 : Scales. Two scales are used to measure the same quantity based on different units of measurement. The measurements in the different scales correspond with linear functions.

Example LAPP.3 : Direct Proportions. Two variables $v$ and $w$ are directly proportional when the ratios between corresponding values of the variables are equal, $v:v' :: w: w'$.Hence , $\frac vw = a \ constant$. The (non-zero) constant is frequently denoted with the letter $c, k$, or $\alpha$. The relation between the two variables can also be expressed with linear functions: $v(w) = c* w$ or $w(v) = c' *v$ where $c' = 1/c$.

Example LAPP.4 : Constant Rates of Change: The rate of change of one variable, $y$ with respect to a second variable, $x$, is defined by the fraction $\frac{change\ in\ y}{change\ in \ x}=r$.

You can use this next dynamic example to solve linear equations like those in Examples LAPP.1 and LAPP.2 visually with a mapping diagram of $f$ (and $g$) and the lines in the graph of $f$ (and $g$).
Example LF.DLAPP.0 Dynamic Visualization of Solving Linear Equations: Graphs, and Mapping Diagrams