Properties of Equations

Lesson Features »

Lesson Priority: High

  • See the definition of what an equation is
  • Know the classifications of equations (open vs closed, true vs false)
  • Define exactly what we mean when we refer to the "solution" of an equation
Lesson Description

Even though you've probably seen the classic "=" before, it's time to take a closer look at exactly what defines an equation, and how equations work.

Practice Problems

Practice problems and worksheet coming soon!


Explaining "Equals"

While I hardly expect that you are seeing and using the "$=$" sign for the first time as you read this sentence, it's possible that your teachers have never formally defined equations. After all, your use and understanding of the symbol is probably correct and good enough to move forward with, without needing a proper dictionary definition. However, there are a few quick properties and ways that we are expected to describe equations that we need to know.Formally, we define an equation as follows.
Define: An EquationAn equation is a mathematical relationship using an equal sign ($=$) to describe two quantities that are the same.We therefore often think of the equal sign as a verb that means "to be the same as", which is appropriate and often convenient.
In short, when we see a math expression like $2/3 = 4/6$, or $5x = 9y$, we are expected to know and to be able to explain that the value or amount of each object in each equation is the same. That is, $2/3$ is the same as $4/6$ and $5x$ is the same as $9y$. Proving and/or solving an equation however relies on a few important properties.

Equation Properties

What are some immediate differences between those two simple example equations $2/3 = 4/6$ and $5x = 9y$? One clear difference is the presence of variables in the second equation. Another is the fact that the first equation is, plainly stated, a true one. $2/3$ IS equal to $4/6$, via the properties of fractions and simplifying fractions. The second equation CAN be true, depending on what value $x$ and $y$ each are. These distinctions, while minute and seemingly inconsequential, are important underlying concepts of equations - ones that we will not often need to state explicitly, but that we will always be expected to be understand and be able to explain if needed.True vs. FalseEquations are often called "number sentences", and just like written language sentences can be true or false, so too can equations.True Sentences$$\mathrm{Australia} \,\, \mathrm{and} \,\, \mathrm{Mexico} \,\, \mathrm{are} \,\, \mathrm{in} \,\, \mathrm{different} \,\, \mathrm{time} \,\, \mathrm{zones.}$$$$9 = 9$$False Sentences$$\mathrm{All} \,\, \mathrm{cats} \,\, \mathrm{have} \,\, \mathrm{hair.}$$$$5=7$$Even if we wish these sentences were true (and I do - Sphynx cats freak me out), these sentences are false. Not all cats have hair, and $5$ is not the same as $7$.Open vs. ClosedEquations that have variables in them are neither definitively true nor false, but can become one or the other if the variable is replaced with a value. For example:$$3x + 1 = 10$$This statement is neither true nor false on its own because $x$ is an unknown variable. Depending on what number we substitute in place of $x$, the equation may become true or false. If $x$ is $3$, then the equation will read$$3x + 1 = 10$$$$\longrightarrow 3(3) + 1 = 10$$$$\longrightarrow 9 + 1 = 10$$$$\longrightarrow 10 = 10 \,\, \checkmark$$If we replace $x$ instead with $5$ then the equation will read$$3x + 1 = 10$$$$\longrightarrow 3(5) + 1 = 10$$$$\longrightarrow 15 + 1 = 10$$$$\longrightarrow 16 = 10$$which is a false equation.Since equations with variables could become either true or false but are not either on their own, we call them open equations. Equations that are entirely numeric are referred to as closed equations.
Define: Equation PropertiesA true equation is one that has two numeric expressions on each side that are or represent the same number.A false equation is one that has two numeric expressions on each side that each represent or simplify to different values.An open equation is one that contains variables, and can become true or false depending on the value of the variable.A closed equation is one that contains only number values or numeric expressions, and is therefore either true or false.Additionally, we say that a solution of an equation is a number that makes an open equation true when it is substituted in place of the variable in that open equation.
It's likely that you're familiar with all of these ideas, even if you never had to give them a name before. And, as you may well already know, when it comes to equations in any math course of study, most of what we will learn is how to find solutions to open equations.

Keeping the Balance

The last property of equations with which you must be familiar is the idea that we can "do stuff" to both sides. Loosely speaking, if we do some math to one side of the equation and then do the same thing to the other side, then the equation will still be valid. For example, if we take one of our earlier sample equations$$5x = 9y$$and we add $3$ to both sides, then the equation is still true.$$5x + 3 = 9y + 3$$If $5x$ and $9y$ are equal, then they have the same value. If they have equal values, then adding 3 to each will also yield two objects of equal value.We'll get our fill of "doing stuff" to both sides throughout hundreds of topics. As we'll see, it's the usual way we work on solving equations.

Put It To The Test

We won't always be heavily tested on these concepts and vocabulary, but we will use these ideas without thinking about them throughout every chapter of Pre-Algebra, Algebra, Calculus, and beyond. If you are taking an early Algebra course, you may encounter quiz or test questions similar to the ones below. Use the definitions in this lesson to answer each question.Labelling EquationsLabel each of the following equations as "open", "closed", "true", or "false". More than one of these may apply.
Example 1$$4 - 1 = 3$$
Show solution
$\blacktriangleright$ This equation is closed because there are only numbers - no variables. Since it is closed, it can be determined to be true or false - in this case $4-1$ is indeed $3$, so this equation is a closed true equation.
Example 2$$6x = 12$$
Show solution
$\blacktriangleright$ This equation is open, and it is neither true nor false.
Example 3$$4 + 11*7 = 105$$
Show solution
$\blacktriangleright$ Without any variables, we know this is a closed equation. Now we just have to figure out if it is true or false.This one requires us to remember how PEMDAS works! If we erroneously added $4$ and $11$ first and then multiplied by $7$, we would get $105$. However this violates PEMDAS, as we must always perform multiplication before addition. Therefore we must multiply $11$ and $7$ first.$$4 + 77 = 105$$$$81 = 105$$This equation is a closed false equation.
Checking SolutionsDetermine whether each proposed value of the variable is a solution to the equation.
Example 4$$2x - 4 = 10$$When $x = 8$
Show solution
$$\blacktriangleright \,\, 2(8) - 4 = 10$$$$16-4 = 10$$$$12=10$$This result is a false equation, so $x=8$ is not a solution to the equation.
Example 5$$9y-9 = 9$$When $y = 2$
Show solution
$$\blacktriangleright \,\, 9(2)-9 = 9$$$$18-9 = 9$$$$9=9$$Because we got a true result when we tried the proposed solution, we know that it truly is a solution to the equation.
Example 6$$-r + 1 = -5$$When $r = 6$
Show solution
$$\blacktriangleright \,\, -(6) + 1 = -5$$$$-5 = -5$$Once again, plugging in and obtaining a true result tells us that what we plugged in is indeed a solution to the equation.
Lesson Takeaways
  • Be able to define an equation using precise language even though you can loosely describe them from our past math experience
  • Know the difference between a true and false equation
  • Understand the difference between an open and closed equation, and for each, the relationship with true / false
  • Describe what it means for a variable value to be a "solution" to an equation using math language
  • Be able to look at equations and know whether open, closed, true, or false applies
  • Know how to check whether or not a proposed solution is correct.

Lesson Metrics

At the top of the lesson, you can see the lesson title, as well as the DNA of Math path to see how it fits into the big picture. You can also see the description of what will be covered in the lesson, the specific objectives that the lesson will cover, and links to the section's practice problems (if available).

Key Lesson Sections

Headlines - Every lesson is subdivided into mini sections that help you go from "no clue" to "pro, dude". If you keep getting lost skimming the lesson, start from scratch, read through, and you'll be set straight super fast.

Examples - there is no better way to learn than by doing. Some examples are instructional, while others are elective (such examples have their solutions hidden).

Perils and Pitfalls - common mistakes to avoid.

Mister Math Makes it Mean - Here's where I show you how teachers like to pour salt in your exam wounds, when applicable. Certain concepts have common ways in which teachers seek to sink your ship, and I've seen it all!

Put It To The Test - Shows you examples of the most common ways in which the concept is tested. Just knowing the concept is a great first step, but understanding the variation in how a concept can be tested will help you maximize your grades!

Lesson Takeaways - A laundry list of things you should be able to do at lesson end. Make sure you have a lock on the whole list!

Special Notes

Definitions and Theorems: These are important rules that govern how a particular topic works. Some of the more important ones will be used again in future lessons, implicitly or explicitly.

Pro-Tip: Knowing these will make your life easier.

Remember! - Remember notes need to be in your head at the peril of losing points on tests.

You Should Know - Somewhat elective information that may give you a broader understanding.

Warning! - Something you should be careful about.

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