Integrated Math 2–3 Honors — What Is an Equation?
Corrected lesson transcript


## 0:00 — What Is an Equation? — Expressions vs. Equations

[0:04] There are two things written on the board, one on the left, one on the right.
[0:08] What do we call the thing on the left?
[0:13] An expression.
[0:14] And what do we call the thing on the right?
[0:16] What is an equation?
[0:33] Anybody know what an equation or what's the definition of an equation is?
[0:37] Take a wild guess.
[0:39] Yeah.
[0:40] Like two expressions brought together by an equal sign.
[0:43] Two expressions brought together by an equal sign
[0:46] So we definitely have one expression another expression then like an equal sign in between okay, we're close
[0:54] Would this be an equation definitely not
[0:58] Will this be an equation?
[1:00] Definitely not.
[1:01] We definitely need an equal sign.
[1:02] And we need an expression on the left and an expression on the right.
[1:06] Here's the actual definition.
[1:08] An equation is a mathematical statement that two expressions are equal.
[1:15] So an equation is a mathematical statement that can be written in many different forms.
[1:22] So why is this an equation?
[1:25] Whoever said it was.
[1:26] Why is it an equation?
[1:27] Because it's saying that 2x minus 3 is equal to 5.
[1:30] Yeah, it's a statement that this expression is equal to this expression.
[1:36] Is this an equation?
[1:38] Yes.
[1:38] Yes.
[1:40] Is this an equation?
[1:42] see everybody's thumb thumbs up yes thumbs down no they're definitely not
[1:55] equal we agree yeah is this a mathematical statement that two
[2:11] Mr. Wittman is 19 years old is that a statement?
[2:19] That true statement
[2:20] No definitely not
[2:25] Okay, so is this a statement that?
[2:29] One expression is equal to another expression. Yes. Yes, this is an equation
[2:35] Okay

## 2:36 — True, False, and Open Equations

[2:37] this is an equation and is the statement true yeah yeah this is called a true
[2:44] equation this is called a false equation
[2:53] Is this equation red?
[2:55] Is it true, false, or neither?
[3:00] It depends on what the value of x is.
[3:02] If x is 3, it would be a true equation.
[3:05] If x is any other number, it would be a false equation.
[3:08] So this is neither true nor false.
[3:10] this is called an open equation. Basically, anytime you see a variable
[3:22] inside an equation it's an open equation.
[3:25] All equations
[3:27] are either
[3:29] true, false, or open.
[3:36] This is very important.
[3:38] Is this an equation?
[3:41] Yes, yes X is an expression X is an expression is a statement the two expressions are equal
[3:49] This contains a variable however. This is the rare case that this is not an open equation. This would be a what?
[3:58] So but most any other time
[4:02] like this equation is no longer true always it's open okay but almost every
[4:10] time except for special cases if an equation contains a variable it's an open equation

## 4:17 — Solving and Equivalent Equations

[4:18] Can
[4:21] Can we solve these?
[4:25] 7th grade?
[4:27] 8th grade?
[4:28] Something like that.
[4:30] What would be the first step?
[4:32] Subtract 5 both sides or add negative 5 both sides.
[4:35] Which property allows me to do that?
[4:39] Either the addition property or the subtraction property.
[4:43] If you add a negative I just call everything the addition property of equality.
[4:49] And then what's the next step?
[4:51] Divide by 2.
[4:53] Okay, let me come back to this.
[5:02] So if the directions were solved, what does solve mean?
[5:11] Make the statement true.
[5:14] Make the statement true.
[5:16] We'll come back to that in a second.
[5:19] Some people said find the answer.
[5:21] What else did we say?
[5:22] Find the unknown value.
[5:24] Okay, it's a possibility.
[5:26] So let's write down the definition.
[5:28] to solve means to find the solution set
[5:47] We'll come back to what solutions that means in a second.
[5:50] What uh, what is the solution to this equation?
[5:54] X equals 3.
[5:56] Uh, it is not x equals 3.
[5:59] It is 3.
[6:02] 3 is the solution. So what's the mean to be a solution?
[6:10] OK, this is very important.
[6:16] Here's the definition.

## 6:16 — Solutions and Solution Sets

[6:17] a solution is a number that when replaced for the variable
[6:40] it makes the equation true
[6:51] OK, so why is the number 3 a solution?
[6:55] Because when we replace it with a variable, we no longer have an open equation, we have
[7:05] a true equation.
[7:07] OK, so a solution is a number that when replaced with a variable, it makes the equation true.
[7:14] Why is the number 7 not a solution?
[7:19] Because when it replaces the variable, it does not make the equation true.
[7:23] Okay.
[7:26] Okay.
[7:28] Solution set, the definition is
[7:33] a solution set
[7:37] is
[7:38] the set of all solutions.
[7:45] So we still haven't solved yet because we haven't found the solution set.
[7:50] Well, we kind of did, but our final answer has to be a solution set.
[7:59] Is x equals 3 the solution?
[8:03] Why is x equals 3 not the solution?
[8:06] It's still an open equation a solution is a number not an equation, okay?
[8:15] So the solution is
[8:20] is three and the solution set you simply take all the solutions there's only one
[8:32] of them here and we put it in these curly brackets there's several ways to
[8:42] that denotes sets in mathematics.
[8:45] If we could list them all, we simply use curly brackets
[8:48] and put them all in there separated by comma.
[8:50] There's only one here.
[8:52] Are these three the same equation?
[8:56] They're definitely not the same.
[9:00] This is a statement that this expression is equal to this expression, which this expression
[9:07] is way different than this expression, and this expression is way different.
[9:12] And this is way different than that one.
[9:14] They're all different equations.
[9:15] However they all have something in common.
[9:17] What do they have in common?
[9:19] The same exact solution set.
[9:21] The same exact solution set.
[9:24] Okay.
[9:25] Like the addition property of equality says you can add the same number to both sides
[9:32] of the equation and the new equation is equivalent.
[9:39] Not the same equations, equivalent.
[9:41] So these are all called equivalent equations.
[9:45] Equivalent equations are equations.
[9:53] that have the same solution set.
[10:06] Another definition.
[10:08] Okay, so we use the addition property of equality to go to here.
[10:12] Here we use the division or multiplication property of equality which says you can divide
[10:18] both numbers by the both sides of the equation by the same number and the new equation, it
[10:26] one's not zero is equivalent which means that is the same solution set okay and
[10:32] once we're here we know the obvious solution is three can we plug in three
[10:36] equals three okay let's try another one
[10:44] The directions say solve.
[10:47] What does it mean to solve?
[10:50] Find the solution set.
[10:52] And what is the solution set?
[10:55] The set of all solutions.
[10:57] The set of all solutions.
[10:58] You take them all and you put them in curly brackets.
[11:02] Okay.
[11:03] And what is the solution?
[11:07] A number that when replacing the variable makes the statement true.
[11:10] Makes the equation true.
[11:12] A number that when replaced for the variable makes the equation true.
[11:17] A number that when replaced for the variable makes the equation true.
[11:22] So solutions, okay first step here is
[11:28] Add eight to both sides
[11:31] Next step
[11:34] Divide by five or multiply by one fifth same thing
[11:38] Is this my solution
[11:42] No, we're getting better, I like this.
[11:45] What's my solution?
[11:47] Twelve fifths, or two point four, something like that.
[11:52] And what is the solution set?
[11:55] Twelve fifths.
[11:56] So the way we say it, we say the set containing 12 fifths.
[12:11] We write down. It's just curly brackets and put the number inside
[12:15] And why is 12 fifth the solution?
[12:18] Because when replace for the variable or plug into the variable it makes the equation true
[12:22] Why is the number zero not a solution?
[12:26] Because when you replace a variable it does not make the equation true.
[12:33] Okay. Let's do another one.

## 12:38 — No Solutions and All Real Numbers

[12:38] Okay, here we got a very well on both sides.
[12:42] So what's our first step?
[12:43] subtract 3x from both sides
[12:50] And we have four equals seven.
[12:54] What does that tell us?
[12:55] It's a false equation.
[12:58] Good.
[13:00] So we use the addition property of equality.
[13:03] We end up with a false equation, always false equation.
[13:08] So what is the solutions?
[13:11] There's none. So is there a solution set?
[13:30] No. I'll leave it.
[13:32] there is.
[13:34] Okay, it's simply
[13:38] this set. The set contains all of its solutions.
[13:44] There are no solutions to this equation, but there is a solution set.
[13:49] So can we even solve this equation?
[13:53] What does it mean to solve?
[13:55] Did we find the solution set?
[13:58] Yes, we can solve this equation.
[14:02] We can't find any solutions but we can solve it.
[14:06] Ok, because by the definition we found the solution set.
[14:11] In fact this is a very important set.
[14:14] Its own name is called the empty set
[14:17] And in fact, it's so important we give it its own symbol
[14:23] Okay
[14:25] It's a circle with a slightly diagonal line through it
[14:31] This is also the empty set
[14:35] You could either write this as your solution set or you could write this as your solution set.
[14:43] Simply an empty set. In fact, it's one of the most important sets in all of mathematics.
[14:49] Is this the empty set?
[14:52] Definitely not, this is not empty.
[14:54] It contains another set.
[14:57] The set containing the empty set is definitely not the empty set.
[15:00] So don't ever write this.
[15:02] Terrible.
[15:04] You can either write this or write this.
[15:07] Don't put one inside the other.
[15:10] Okay? Okay. Let's try a new equation.
[15:16] This has lots of variables, what do we get first?
[15:19] I like my terms like that
[15:24] And now what
[15:26] You could either stop here
[15:29] This is always equal to this
[15:32] If you want to we could subtract 4x
[15:35] we get this equation
[15:38] and it doesn't matter if we do that or not
[15:42] and what do we end up with?
[15:46] an always true equation
[15:48] so what are the solutions?
[15:51] Every single number we know is a solution. And what is the biggest set of numbers we
[15:57] know? All real numbers.
[16:03] Why is every real number a solution? Because when you plug it in for the variable it makes
[16:08] the variables in this case makes the equation true.
[16:14] Okay, what is the solution set?
[16:18] Okay, the solution set the set of real numbers.
[16:22] Okay.
[16:24] And there's lots of different ways we denote the set of real numbers
[16:30] One way is we write it out in words like that
[16:34] Another way is
[16:37] It looks like this have we seen that before
[16:41] Okay
[16:43] That is simply the letter R. But in boldface
[16:49] That's not IR it's bold face are
[16:54] Right when you're in Microsoft Word or whatever how do you make something bold?
[16:59] We highlight it and press bold
[17:03] Okay
[17:04] Before computers you couldn't do that. You had to make it bold by hand and so back who knows how long
[17:12] Okay, they made it bold by
[17:15] making the left side
[17:18] Put a line there, okay, that's simply bold face are
[17:24] It's just that is the one way we denote the set of real numbers. We're gonna learn another way. Some of you might know this
[17:31] we do this all the numbers between negative infinity and positive infinity
[17:38] okay we'll typically do this way this is called interval notation
[17:43] how many of you guys learned interval notation last year
[17:47] and not too many we'll learn it uh beginning next week i think
[17:55] Okay, so some equations have one solution, some equations have no solutions, and very
[18:05] similar equation all of a sudden we have an infinite number of solutions.
[18:09] Alright, either way, any equation we can solve even if there's no solutions.

## 18:17 — Equations in Two Variables

[18:17] Is this an equation?
[18:19] Why is it an equation?
[18:21] It's a statement that two expressions are equal.
[18:24] As long as you see an equal sign and some expression on the left and some expression
[18:27] on the right, it's an equation.
[18:29] Anybody know a solution to this?
[18:33] Just like say like x can go y and y can go like 5.
[18:40] Close. So this is fundamentally different because there's two variables.
[18:47] So we actually have to alter our definition.
[18:54] Where's our definition of equation?
[18:57] Here it is.
[18:58] An equation is a mathematical statement that...
[19:01] Sorry.
[19:03] Definition of a solution.
[19:04] A solution is a number that when replaced for the variable,
[19:10] this is really the only definition for an equation in one variable.
[19:15] Okay, when we have an equation in two variables, it's a different definition. Let's write down
[19:25] definition. A solution, equation, and two variables. It's not a number. We
[19:53] need two different numbers okay and right for example one comma five
[20:03] somebody said or one for the X five for the Y is the solution so we had to know
[20:10] which number goes to which variable and the way we denote that is an ordered
[20:16] pair. So a solution to the equation in two variables is an ordered pair such
[20:28] that when the numbers are replaced for their respective variables it makes the
[20:56] equation true. So it's not a number, it's two numbers. We should call it ordered pair.
[21:12] Such that when the numbers are replaced for the respective variables, the x for the x
[21:17] my life or why it makes the equation true. So 1 comma 5 is a solution. Somebody give
[21:31] me another solution zero comma three
[21:40] well okay interesting I like that one half four three comma nine 100 comma
[21:57] 203; negative 10, negative 17 and we could keep going how many solutions
[22:16] are there to this equation? Infinite. Okay there's no way we're gonna write down all
[22:28] the solutions so we're not going to do a curly bracket so we can't write them all
[22:36] plan. So how are we going to write down the solution set?
[22:47] Any guesses? Okay, so I have a question.
[22:54] So what would I write?
[22:57] So all real numbers?
[23:01] And the range is?
[23:05] All real numbers.
[23:08] So does that mean every ordered pair is a solution?
[23:12] No.
[23:13] It's definitely not.
[23:16] 0, 10 is not a solution.
[23:19] So we can't just write down the domain range.
[23:22] We'll talk about what domain range means in a minute.
[23:24] How am I going to do this?
[23:27] Because we know lots of numbers that are not solutions.
[23:29] Sorry, lots of ordered pairs. This definitely is a linear equation. Okay, in
[23:49] fact, you've written down the solution set to this many, many, many times. How do
[23:56] we do it in eighth grade? We graphed the solution set. Okay. We draw a coordinate plane.
[24:18] So when you graph linear equations in two variables in eighth grade, and you probably
[24:24] did a bunch last year too, you're really just graphing the solution set.
[24:29] All the ordered pairs that were replaced for the variable make the equation true.
[24:35] There's an
[24:36] This line is simply an infinite number of ordered pairs
[24:42] Every single ordered pair on that line is a solution all of them together is the solution set
[24:49] Okay
[24:51] We could have graphed the solutions for these these are actually also called linear equations
[24:57] but only in one variable. We could graph it. There's only one variable we need only
[25:02] one number line. We could put a dot right here on 12 fifths and we graph the
[25:14] solution set. We put a dot on every solution. We don't do that because it's
[25:18] pointless. I mean it's a lot easier to do this. Okay we could have graphed this
[25:26] solution set here and we put a dot on every single solution which is all written numbers
[25:36] but we're gonna do that okay but we do graph the solution set to linear equations in two variables
[25:45] it's it's really the only way to write them all down there are some other notations that we don't
[25:52] to worry about that right now okay this tells you what is a solution what is not
[25:58] a solution okay so did we solve this equation yeah why do we solve it found
[26:21] the solution set. Okay, this is very important. We're always either going to start with an
[26:28] expression this class or an equation. Okay, this very first equation, sorry, very first
[26:38] Expression I wrote here. Can we solve this?
[26:42] Why can we not solve
[26:44] It's
[26:45] it's not an equation to solve needs to find the solution set a
[26:51] solution set that all solutions a solutions a number that replace for the variable makes the equation true
[26:58] There's no equation
[27:01] Expressions are neither true or false
[27:04] Okay, just is what it is
[27:06] Okay, you cannot solve this you can't solve this an equation
[27:11] okay, so when we
[27:13] When you look at something around the board it's either going to be an expression or an equation
[27:19] We do totally different stuff
[27:21] Almost always we solve these
[27:23] Expressions we can do we can manipulate them
[27:26] You could rewrite them.
[27:31] You could factor out a number.
[27:37] You can change it, manipulate it,
[27:39] simplify it, complexify it,
[27:41] but you cannot solve it. Very important.

## 27:43 — Relations and Their Representations

[27:44] A new definition.
[27:48] The definition of a relation is simply a set of ordered pairs.
[27:53] It's called a relation.
[27:56] Is this a relation?
[27:59] Yes.
[28:00] Yes.
[28:01] Yes, why?
[28:03] Because it's a set of ordered pairs.
[28:05] Curly brackets mean it's a set.
[28:07] and the inside separated by commas are?
[28:11] Ordered pairs.
[28:11] Ordered pairs, this set of order of pairs.
[28:14] Okay.
[28:16] This has a very clear pattern.
[28:17] Well, it should be clear.
[28:21] 1, 3, 2, 5, 3, 7, what's the next one?
[28:25] 4, 9.
[28:26] 4, 9, so on and so forth.
[28:28] If there's a clear pattern, you just put dot dot dot, the ellipses.
[28:32] Okay, so here this is a relation. Why is this a relation?
[28:37] Why is this a relation?
[28:40] Because it's a set of ordered pairs. That's right
[28:44] Because there's the brackets and there's everything inside is ordered. This has an infinite of ordered pairs in it
[28:50] Okay
[28:51] Okay, there are three ways to denote relations.
[29:03] First of all, you list the ordered pairs.
[29:12] Just like we did here and here.
[29:15] Okay. It could be in a table too. Doesn't necessarily have to look exactly...
[29:22] Okay. You can list them like that in the correct... This is called roster
[29:28] notation you put them in a table you can go 1 5 negative 4 and put arrows 2 ok
[29:44] doesn't really matter as long as you listing your ordered pairs. The second way what
[29:51] What else can we do with ordered pairs?
[29:53] Just like we did up here, we graphed them.
[29:57] We graphed the relation.
[30:00] Thank ya.
[30:02] Okay, it could be like this or like we did earlier.
[30:05] We can have an infinite number of ordered pairs.
[30:09] That's typically when we graph a relation.
[30:11] When there's an infinite number of them.
[30:14] And then what's the third way?
[30:17] Do you want to take a guess what the third way is?
[30:20] This
[30:23] Was the solution set this guy, right
[30:28] Okay, so the third way is we write down an equation
[30:33] Okay
[30:35] However, an equation is not a set of ordered pairs.
[30:42] Okay, so if we write down y equals 2x plus 3, this is not a set of ordered pairs.
[30:51] But, what about this is a set of ordered pairs?
[30:57] The solution set.
[30:59] So in equation, it's the solution set.
[31:04] OK.
[31:06] It's the solution set that is the set of ordered pairs. It is the relation. So we'll call this a relation
[31:14] Technically, it's not the solution set that this is the equation
[31:20] If the solution set is the set of ordered pairs, okay, so those are three ways we denote relations

## 31:28 — Inequalities

[31:28] Is this an equation?
[31:34] Wait, will?
[31:36] Sometimes it's a yes.
[31:38] Definitely not. Is that an equal sign?
[31:41] No.
[31:42] No, definitely not an equation. What do we call this thing?
[31:47] An inequality. Okay.
[31:50] Can we solve inequalities? What's the need to solve?
[31:55] Find the solution set.
[31:57] Okay, it needs to find the solution set.
[32:00] What's the definition of a solution to an inequality?
[32:04] Actually let's go back.
[32:05] what is an inequality what is it
[32:24] So, what is the definition of an inequality?
[32:27] Umm, close. Not, well, not really.
[32:31] A question?
[32:33] Sort of. It's almost the same definition as equation.
[32:39] Or is equation?
[32:40] A mathematical statement that you use questions or a less than variable?
[32:46] Okay, so instead of equations, it's going to be an inequality.
[32:51] We'll write this down in a second.
[32:52] It's a mathematical statement that one expression is either less than, greater than, less than
[32:59] or equal to, greater than or equal to another expression.
[33:01] Let me write that down.
[33:02] community
[33:03] and inequality
[33:07] is a mathematical statement
[33:09] that one expression
[33:12] is less than
[33:17] or greater than or less than or equal to or greater than or equal to
[33:24] another I could have wrote the words less than or greater than or less than or
[33:54] this is it gonna be a number or an ordered pair a number what because only one
[34:07] variable okay how many solutions does this have okay is zero a solution yes is
[34:32] Is 3 a solution? Yeah. Is 4 a solution? Yes. Is 4.5 a solution? Yes. Is 5 a solution? Yes. Yes. Why is 5 a solution?
[34:45] Because it has a variable 2. When replaced for the variable, it makes the inequality true.
[34:52] Because of the or equal to, this is true. Okay. So it definitely has an infinite number of solutions.
[35:00] We can write them in something called interval notation.
[35:03] We'll learn that probably Monday.
[35:06] Right now we can just graph the solution set.
[35:09] Since there's only one variable, we need only one number line.
[35:14] Okay, and you simply put a dot on every single solution.
[35:21] You did this last year, correct?
[35:24] Solved inequalities.
[35:25] You typically graph a solution set.
[35:30] Let's put negative nine.
[35:34] What was our first step?
[35:37] and then divide by 2 do I have to flip the inequality sign? No. No. Because we're not
[35:56] No. No. All three of these are called equivalent inequalities.
[36:03] Okay.
[36:06] So we're going to graph the solution set.
[36:10] We've got to put a dot on every single solution.
[36:14] Is the number 3 a solution?
[36:16] Sorry, is the number negative 3 a solution?
[36:18] No.
[36:19] No.
[36:20] Okay.
[36:22] We have to just remove one tiny point.
[36:26] Okay, it's very hard.
[36:27] If you pull that one tiny point,
[36:29] you'll never see that it's gone.
[36:30] so we exaggerated it make a big open circle even though okay again there's

## 36:43 — Key Takeaways

[36:43] gonna be a better way to do the solution set but for right now we're just gonna
[36:48] graph the solution set okay all these definitions are very important what's it
[36:56] Equation means we could make the solve with a solution set with a solution
[37:02] Put our equivalent equation or equation true false and open
[37:08] We got to memorize these
[37:10] Know them by heart real quick
[37:13] We will have a quiz probably in the next week on the definitions