High School: Algebra
Reasoning with Equations and Inequalities HSA-REI.D.10
10. Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line).
Students should understand that equations with two variables can be represented graphically. The shape that results on the coordinate plane is a visual representation of all the solutions to that equation.
What does that mean? It means we're not just pulling rabbits out of hats! The equations actually mean something visually.
An equation with two variables can be anything from y = x to x2 + y2 = 4 to 19x13 = y. Some are simpler than others, of course, but they all have an x and a y.
That means instead of having an equation with one variable (and therefore one solution), we can have many different solutions. Graphically, we can represent these solutions by drawing a curve or line through all the pairs of solutions (one for x and one for y) that work for that particular equation.
Let's take the equation 7x – 18 = y and see how we can represent this graphically.
How do we prove to a student that indeed, a line of a two-variable equation, when graphed, shows all of the solutions? Let's show them how to pull that rabbit out of the hat themselves.
Since any two points define a line, all we need to do is input two values for x and see what the output y values are. We'll pick three points just to be sure our graph is a line and not some weird curve.
Let's pick the numbers -1, 0, and 3 for x. Plugging in the numbers for x into our equation 7x – 18 = y gives us -25, -18, and 3 for the y values. So the points in our graph become (-1, -25), (0, -18), and (3, 3). If we graph these on the x-y coordinate plane, we'll have this:
If your students don't believe you, prove to them that the equation and graph correspond to one another. Take a point on the line that is easily identifiable, say (2, -4), and plug the values into the equation. If we do that, we'll have -4 = 7(2) – 18, which simplifies to -4 = -4. That way, students will be sure that points on the line or curve are valid solutions to the equation, and vice versa.
But don't stop there. It's also important to prove the opposite. For example, the coordinate (4, 1), which is not on the line, is also not a solution to our equation. If we plug in the coordinates, we can confirm this: 1 = 7(4) – 18 is false because 1 ≠ 10. This means (4, 1) isn't a solution to our equation and not a point on the line.
Now you can pat yourself on the back and prove to students that teachers aren't just up to some magic tricks. Everything in math pretty much works as it's supposed to.
This method can be applied to two-variable equations of higher orders. The generic shapes of these equations (such as quadratics making a parabola) should be known and associated with each other already. Otherwise, students will need to graph several points before verifying the graph that corresponds with the particular equation.