Why a Graph Is the Set of All Solutions

In this lesson you will: Decide by substitution whether an ordered pair is a solution of a two-variable equation; Explain that the graph shows every solution and nothing that is not one; Find the missing coordinate of a point known to lie on a line.

A two-variable equation is solved by pairs

The equation 3x-12=0 has one variable, and its solution is a number: 4. The equation 3x-4y=12 has two, so a single number cannot satisfy it. A solution here is an ordered pair that makes the two sides equal.

(4,0) works, since 12-0=12. So does (0,-3), since 0+12=12. So does (8,3), and so does (2,-1.5). There is no shortage of them: infinitely many pairs satisfy this one equation.

The line is the solution set — and that runs both ways

A.REI.10 makes two claims at once, and students are usually taught only the first. One: every point on the graph is a solution of the equation. Two: every solution of the equation appears on the graph — none is left off.

The second claim is what makes a graph an answer rather than an illustration. If you want to know whether some pair is a solution, the line already contains the verdict: on the line means yes, off the line means no.

Which is why a point that misses the line by a hair is not a solution. It is simply a pair that fails the equation, drawn near one that passes.

Running the test backwards

If you are told a point lies on a line, you have been told its coordinates satisfy the equation. That turns the equation into a one-variable equation you can solve for whichever coordinate is missing.

Suppose (a,-4) is on 2x-5y=34. Substituting gives 2a-5(-4)=34, so 2a+20=34 and a=7. The point is (7,-4).

Common mistake — Deciding from the picture instead of the arithmetic

Asked whether (5, 0.8) is a solution of 3x-4y=12, a student finds the point on a printed graph, sees it sitting on the line, and answers yes.

Substituting: 3(5)-4(0.8)=15-3.2=11.8, and 11.8 != 12. The point is not a solution.

The pair that really is on the line at x=5 is (5, 0.75), since 15-3=12. The gap between 0.8 and 0.75 is about a twentieth of a unit — thinner than the printed line, and invisible at any sane scale.

A second version of this error checks only one coordinate: \"x=5 gives 15, and 15 is bigger than 12, so yes.\" Half a substitution is not a check.

How to catch it: Compute the left side and the right side separately, then compare the two numbers. If they are not equal, the pair is not a solution, no matter how convincing the picture was.