Equations

When you see an equation, what purpose is it actually serving? E.g., "$x = 5$"

We can read that as “x equals 5,” but what does that mean? What is it telling you?

Let’s say it a different way:

• ”$x$ is the same as $5$”
• ”$5$ can be substituted for $x$”

Equations are symmetrical, that means we can reverse them and read them in either direction without changing their meaning.

$x = 5 \implies 5 = x$

Stated: “5 is the same as $x$” or ”$x$ can be substituted for 5”.

The “Students Must Be Solving” Trap

Misconception: I think most students see equations as unanswered questions that need to be solved.

And it’s often the case that there is something we want to find when we’re using an equation.

The problem with this notion is that it obscures what equations actually are.

Equations are simply one of many tools that we can use to find values in various scenarios.

Try not to think of equations as something to solve.

Rather, solving (assigning values to variables to make some statement true, usually) is something we can use equations to do.

Correction: Equations are merely declarations of equality.

They’re just statements. The math equivalent of a sentence, something we say to relativize quantities to one another.

Hopefully that’s quite obvious, though maybe you’ve never said it or thought about it explicitly in that way.

How have we used equations so far?

1. Function definitions

   We have declared function rules (e.g., "$f(x) = x^2 + 3x - 10$") using equations.
   This doesn’t necessarily have to be written as an equation. We could instead say:

   “Take a value, which we will call x, square it, and add that square to 3 times x. Then, subtract 10 from that sum,” and we have defined our function through prose. Perhaps, though, from this example, you can see the usefulness of both function notation and the conventional order of operations. It’s concise, unambiguous, and we can do math with it.

2. Finding x-intercepts

   We have also used equations to help us find x-intercepts. We know that the output (y-value) of a function at its x-intercept(s) will be 0. However, we might not know what the input values are that result in an output of 0, so we set up an equation and solve it to find them:

   $0 = x^2 + 3x - 10$

   Solving this equation for x allows us to find the x-intercept(s) of $f(x)$ as defined above.

   What is the above equation saying?

   “0 is what we get when we square x, add that to 3 times x, and then subtract 10.”

   Then, our task becomes to solve for x. The equation isn’t asking a question; we’re asking the question, “When does $f(x)$ cross the x-axis?” and using the equation to find the answer.

3. Can you think of any other ways we’ve used equations?

Solving equations:

It is helpful to think of solving an equation as manipulating the equation while preserving equality. Equality is preserved when we perform operations by performing them simultaneously on both sides of the equal sign.

We’ve discussed following an inverse order of operations and performing inverse operations/functions to manipulate equations.

Let’s look at another way to solve simple equations visually that will develop our notion of equality.