Common Pairs

Your journey through mathematics has brought you to a major change in your thinking. You have been working with individual numbers for years and years and years. Numbers have been the fundamental object in mathematics. Functions bring a totally different viewpoint. Our new basic mathematical tool is expanding from a number to a function. Our new basic thought is about huge collections of pairs of numbers. Our new analysis is moving from how numbers combine and compare to how functions combine and compare.

We are learning how to investigate functions. Given a function, we would like to know its domain and range, zeros, discontinuities, singularities, where it is increasing or decreasing, what are the maximum and minum values, and what its graph looks like.

We would also like to compare functions. For instance, what pairs functions might have in common. Let’s try this for linear functions.

Intersections

Given two linear functions (1) do they share any pairs? (2) what are the common pairs?

If $f$ and $g$ are two linear functions and they shared the pair $(a, b)$, then we would have $f(a)=b$ and $g(a)=b$. Their graphs would share a point. The lines would intersect. Since each linear function corresponds to exactly one line, it might be benefitial to think geometrically.

How can two lines intersect?

One Point

Two lines can interesect in exactly one point.

One point of intersection is probably the usual expectation.

The other two possibilities are more uncommon.

No Intersection

Parallel lines would not intersect.

Distinct linear functions with the same rate of chage cannot share any pairs.

Everywhere

Two lines that were in fact the same line, would intersect at every point, because they are the same line.

While this is completely obvious when examining graphs, it might not be so obvious during an algebraic investigation.

It might not be immediate clear that $L(k) = \frac {4}{3}k - \frac {8}{5}$ and $p(h) = \frac {4(5h-6)}{15}$ are in fact the same linear function and share every pair.

Proof

Linear functions can share no pairs, exactly $1$ pair, or all of their pairs. Is that it? Our intuition tells us that the graph of a line cannot turn around and have a second intersection point. But, that is far from a convincing argument. Perhaps the graphs do not show enough and our intuition is wrong.

How do we convince people that this is true, without saying “trust me”?

Graphing and Geometry are excellent tools for believing, but when you need an argument that accounts for EVERYTHING, then you need Algebra. Algebra helps us make sure we have accounted for EVERYTHING.

Reasoning that accounts for everything, are called proofs.

As a consequence, if you know you have two linear functions and you know they share two points, then they must share all of their points and they must be the same function.

This is why we only need two points to draw a line.

Systems of Linear Equations

The thinking above is part of a larger idea.

While we are currently considering two variables, a general linear equation can have any number of variables.

$$a_n \, x_n + a_{n-1} \, x_{n-1} + \cdots + a_2 \, x_2 + a_1 \, x_1 + a_0 = 0$$

We might collect some linear equations and group them together. This collection is called a system of linear equations. And, as above, we might look for common solutions. The process of looking for common solutions is called solving the system.

Systems are commonly written together with the equal signs aligned.

For example, the system might include two equations with two variables.

For example, the system might include two equations with three variables.

For example, the system might include three equations with three variables.

Or, some other combination. A system of linear equations can group any number of linear equations with any number of variables. The study of these types of systems is called Linear Algebra.

For us, solving the system - identifying the common solutions - always uses the same thinking - EQUALITY.