tags:
  - algebra-1
  - secondary-level

Graphing equations and inequalities in two variables

Ordered pairs and the coordinate plane

A linear equation in two variables is a Linear equations with two variable. The solution/s are written as an ordered pair.
An ordered pair is written (a, b) where a is the x coordinate and b is the y coordinate (see graphing).
We tend to find a given number of ordered pair solutions so we can graph the result.
To create a table we plug in one value and then solve for the other.
Cartesian co-ord plane has 4 quadrants, labelled I to IV starting top right and going anti-clockwise.
- I is (+, +)
- II is (-, +)
- III is (-, -)
- IV is (+, -)
- Origin is (0, 0)

II	I
III	IV

Forms of the line

Why and how to use slope-intercept form.
- y = mx + b
- Difficult to find a table of ordered pairs, so we set x=0, get the y-intercept and the slope, which is enough to graph the line.
Why and how to use point-slope form.
- y - b = m(x - a)
- When we know a slope and one point (a, b)
Standard form
- ax + by = c or similar (defs. vary)
- Lets us find x & y intercepts then graph
- We can convert to std form by getting x any onto one side of the equation

Graphing linear equalities and inequalities

Solve for y (if an inequality remember when to reverse the sign, if just x or y it's a vertical or horizontal line)
Put into slope-intercept form y = mx + b
For an equality simply plot y-intercept (0, b) and another point or two on the slope.
For an inequality make a dashed line for strict and a solid line for ≤ ≥
Shade or indicate the solution space, above the line for > ≥ below for < ≤

Help from friends

Infinite number of solutions?

Greene Math: When we have a linear equation in two variables, there are an infinite number of solutions. This means there are an unlimited number of (x,y) ordered pairs that will satisfy the equation.

Marsh: Counting all the real numbers that fit as values? Including all the decimals? Up to now all the answers to test problems have been been integers or fractions (rational numbers).

D--p: yes

If you have a 2-variable linear equation, you can always rearrange it to look like y=a*x+b for some a,b . Then, you can choose any real number, rational or irrational, for x, slot it into that formula, and get a corresponding value for y.

So there’s an infinite (in the sense of, one for every real number) number of possibilities for x; for each of those there’s a corresponding real y, so there is an infinite number of (x,y) pairs

You can certainly restrict yourself to only integer or rational values, and maybe that’s the right thing to do (eg if the linear equation in question relates discrete quantities like “number of people”) but the linear equation itself will happily work with any real values

(In fact you can generalise it to work with complex values or vector values or even weirder stuff like “elements of the set of functions defined on a specified range” - when you do that you’re into the field of linear algebra, but it all comes from the same root)

Or there’s the geometric interpretation: every 2-variable linear equation describes a line in 2-d space - the line of the graph of y=a*x+b as obtained above. Each point on that line corresponds to an ordered (x,y) pair. Just as there’s an infinite number of points on the line, there’s an infinite number of such ordered pairs.

Similarly: it’s reasonable in certain contexts to draw the line on squared paper and only pay attention to where it intersects the grid, corresponding to integer x and y. But you don’t necessarily have to do that; the line itself is infinite in extent and infinitely divisible.

Shading inequalities on paper

Marsh: Quick question maths. When graphing an inequality, what's the pen and paper way to shade the right region? Like, if you were in an exam?

H----e: I recall teachers saying it doesn't matter but you do have to provide a key and get it the right way round there.

D--p: I was taught that you hatched the line on the opposite side to that indicated by the inequality, because if you have multiple inequalities then it gets to be a real mess otherwise (Image: Friend's graph of 3 inequalities on one graph) Doing linear programming problems is a right pain otherwise

There’s a whole little subfield called “linear programming” or “linear optimization” where you’re given a set of constraints like this and you have to find the region where they’re all satisfied (the triangle above), and then optimize something within that region