What similarities and differences are there between functions and linear equations? Are all linear equations functions? Is there an instance when a linear equation is not a function?

A function is a rule that explains how one variable can be computed from one or more other variables. A linear equation of the form y = mx + b is a function.

If you have an equation without ONLY one variable on one side, it is technically not a function, in my opinion.

b = mx - y and m = y/x are not explicit functions as written, if m and b are constants, but can be rewritten as linear functions. Those might be called implicit functions.

I think one get lost in semantics with questions like this. I wonder whether it really helps in teaching math.

With a function, for every X there is only one Y. With linear equations, for every X, there can be more than one Y.

Functions and linear equations are closely related, but they do have some similarities and differences.

Similarities:
1. Both functions and linear equations are mathematical representations of relationships between variables.
2. They both have inputs (independent variables) and outputs (dependent variables).
3. Both can be expressed using mathematical notation and symbols.

Differences:
1. Functions are more general and flexible than linear equations. A function can represent any type of relationship between variables, while a linear equation specifically represents a linear relationship.
2. A linear equation is a type of function that follows a straight line on a graph. It has the form y = mx + b, where m represents the slope of the line and b represents the y-intercept.
3. Functions can take on various forms, including linear equations, but they can also be quadratic, exponential, logarithmic, etc.

While all linear equations can be represented as functions, not all functions are linear equations. For instance, a quadratic function like y = x^2 is not a linear equation. In a quadratic function, the relationship between variables is not a straight line but a curve. Similarly, exponential and logarithmic functions are not linear equations.

To determine if a linear equation is a function, we can use the vertical line test. If any vertical line intersects the graph of the equation at more than one point, then it is not a function. However, if every vertical line intersects the graph at most once, then it is a function. Since a linear equation represents a straight line, it always passes the vertical line test and is always a function.