Algebra Basics: What Is Algebra? - Math Antics - Free Educational videos for Students in K-12 | Lumos Learning

Algebra Basics: What Is Algebra? - Math Antics - Free Educational videos for Students in k-12


Algebra Basics: What Is Algebra? - Math Antics - By Lumos Learning



Transcript
00:03 Uh huh . Hi , this is Rob . Welcome
00:07 to mathematics . In this lesson , we're going to
00:09 learn some really important things about a whole branch of
00:13 math called algebra . The first thing you need to
00:16 know is that algebra is a lot like arithmetic .
00:19 It follows all the rules of arithmetic , and it
00:21 uses the same four main operations that arithmetic is built
00:25 on . Addition , subtraction , multiplication and division .
00:30 But algebra introduces a new element . The element of
00:33 the unknown . When you were learning arithmetic , the
00:37 only thing that was ever unknown was the answer to
00:39 the problem . For example , you might have the
00:42 problem . One plus two equals What ? The answer
00:46 isn't known until you go ahead and do the arithmetic
00:49 . Now , the important thing about algebra is that
00:51 when we don't know what a number is , yet
00:53 we use a symbol in its place . That symbol
00:56 is usually just any letter of the alphabet . A
00:59 really popular letter to choose is the letter X .
01:02 So in arithmetic , we would just leave the problem
01:05 like this one plus two equals blank . And we
01:08 had right in the answer when we did the addition
01:11 . But in algebra we would write it like this
01:14 one plus two equals X . The X is a
01:17 placeholder that stands for the number that we don't know
01:20 yet . What we have here is a very basic
01:23 algebraic equation . An equation is just a mathematical statement
01:28 that two things are equal . An equation says the
01:31 things on this side of the equal sign , have
01:33 the same value as the things on the other side
01:35 of the equal sign in this case are equations telling
01:39 us that the known values on this side one plus
01:42 two are equal to what's on the other side ,
01:44 which happens to be the unknown value that we're calling
01:47 X . One of the main goals in algebra is
01:50 to figure out what the unknown values and equations are
01:53 , and when you do that , it's called solving
01:55 the equations . In this equation , it's pretty easy
01:59 to see that the unknown value is just three .
02:01 All you have to do is actually add the one
02:03 and two together on this side of the equation ,
02:06 and it turns into three equals X , which is
02:08 the same as X equals three . So now we
02:11 know what X is . It's just three that almost
02:14 seems too easy , doesn't it . And that's why
02:17 , in algebra , you're usually given an equation in
02:19 a more complicated . For like this , X minus
02:22 two equals one . This is exactly the same equation
02:26 as one plus two equals X , but it's been
02:28 rearranged so that it's not quite as easy to tell
02:31 what X is so in algebra . Solving equations is
02:35 a lot like a game where you're given mixed up
02:37 complicated equations , and it's your job to simplify them
02:40 and rearrange them until they're nice . Simple equations like
02:43 X equals three , where it's easy to tell what
02:45 the unknown values are . We're going to learn a
02:48 lot more about how to actually do that , how
02:50 to solve equations in the next several videos . But
02:53 for now , let's learn some important rules about how
02:56 symbols can and can't be used in algebraic equations .
02:59 The first rule you need to know is that the
03:02 same symbol or letter can be used in different algebra
03:05 problems to stand for different unknown values . For example
03:09 , in the problem , we just solved the letter
03:11 X was used to stand for the number three right
03:14 , but X could stand for a different number in
03:16 a different problem . Like if someone asks us to
03:18 solve the equation . Five plus X equals 10 .
03:22 In order for the two sides of this equation ,
03:24 to be equal , X must have the value five
03:26 in this problem , because five plus five equals 10
03:30 . So X or any other symbol can stand for
03:33 different values in different problems . That's okay , but
03:36 what's not okay is for assemble to stand for different
03:39 values in the same problem at the same time .
03:42 For example , what if you had the equation X
03:45 plus X equals 10 ? This equation says that if
03:49 we add X to X will get 10 . And
03:52 there's a lot of different numbers that we could add
03:53 together to get 10 like six and four . But
03:57 if we had the first X stand for six and
04:00 the second X stand for four , then X would
04:02 stand for two different values at the same time .
04:05 And things could get really confusing . Uh , if
04:08 you wanted symbols to stand for two different numbers at
04:10 the same time , you need to use two different
04:13 symbols , like X and y so in algebra ,
04:16 whenever you see the same symbol repeated more than once
04:19 in an equation , it's representing the same unknown value
04:23 . Like if you see a really complicated algebraic equation
04:26 like this , where X is repeated a lot of
04:29 different times , all those exes stand for the same
04:33 value , and it will be your job to figure
04:35 out what that value is . Okay , so for
04:38 any particular equation , we can't use the same letter
04:42 to represent two different numbers at the same time .
04:44 What about the other way around ? Could we use
04:47 two different letters to represent the same number ? Yes
04:51 . And here's an example of that . Let's say
04:54 you have the equation . A plus B equals two
04:57 . What could A and B stand for ? So
04:59 that the equation is true ? Well , if a
05:02 was zero and B was to , then the equation
05:05 would be true . Or we could switch them around
05:08 if they was to and be with zero . The
05:10 equation would also be true , but there's another possibility
05:14 . If a was one and B was also one
05:17 that would make the equation true , right ? So
05:19 even though a and B are different symbols and would
05:23 usually be used to represent different numbers , there are
05:26 times when they might happen to represent the same number
05:29 . Oh , and this problem can help us understand
05:32 something very important about how symbols are used in algebra
05:36 . Did you notice that there were different possible solutions
05:38 for this equation ? In other words , be could
05:41 have the value 01 or two , depending on what
05:44 the value of aid was . If a is zero
05:48 , then be must be , too if a is
05:50 one , then be must be one . And if
05:52 a is to then be must be zero . You
05:56 can't have two different values at the same time ,
05:58 but its value can change over time if the value
06:01 of a changes in algebra bees what's called a variable
06:05 because its value can vary or change . In fact
06:08 , in this equation , both A and B are
06:10 variables because their values will change depending on the value
06:13 of each other . Actually , it's really common in
06:17 algebra to refer to any letter as a variable ,
06:20 since letters can stand for different values and different problems
06:23 . But at math , antics will usually just use
06:25 the word variable when we're talking about values that can
06:28 change or vary in the same problem . Alright ,
06:32 so far we've learned that algebra is a lot like
06:34 arithmetic , but that it includes unknown values and variables
06:38 that we can solve for in equations . There's one
06:41 other really important thing that I want to teach you
06:44 that will help you understand what's going on in a
06:46 lot of algebra problems , and it has to do
06:49 with multiplication . Here are the four basic arithmetic operations
06:54 addition , subtraction , multiplication and division , although in
06:58 algebra you'll usually see division written infraction form like this
07:03 in arithmetic . All four operations have the same status
07:06 , but in algebra , multiplication get some special treatment
07:11 in algebra . Multiplication is the default operation . That
07:15 means if no other arithmetic operation is shown between two
07:18 symbols , then you can just assume that they're being
07:21 multiplied . The multiplication is implied . For example ,
07:26 instead of writing a Times B , you can leave
07:29 out the time symbol and just write a B .
07:31 Since no operation is shown between these two symbols ,
07:35 you know that you're supposed to multiply A and B
07:38 . Of course , you can't actually multiply a and
07:40 B until you figure out what numbers they stand for
07:43 . The advantage of this rule about multiplication is that
07:46 it makes many algebraic equations less cluttered and easier to
07:50 write down . For example , instead of this a
07:53 Times B plus C Times d equals 10 . You
07:57 can just write a B plus C d equals 10
08:01 . You can also use this shorthand when you're multiplying
08:03 a variable and unknown number like two X , which
08:07 means the same thing as two times X or three
08:10 y , which means the same thing as three times
08:13 why , since the symbol and the numbers are right
08:16 next to each other , the multiplication is implied .
08:18 You don't have to write it down . Finally ,
08:21 some good news . Now I never have to write
08:23 down that pesky multiplication symbol again . Oh , yeah
08:27 , uh , not so fast . There are some
08:30 cases in algebra where you still need to use a
08:33 multiplication symbol . For example . What if you want
08:36 to show two times five . If you just get
08:38 rid of the time simple . And put the two
08:40 right next to the five , it's going to look
08:42 like the two digit number 25 which is not the
08:45 same as two times five . So whenever you need
08:49 to show multiplication between two known numbers , you still
08:52 have to use the time symbol unless you use parentheses
08:55 instead . But aren't parentheses used to show grouping and
08:59 meth . How can you use that to show multiplication
09:03 ? Ah , that's a good question . Parentheses are
09:06 used to group things , But whenever you put two
09:08 groups right next to each other with no operation between
09:11 them , guess what operations implied ? Yep , Multiplication
09:16 , for example . If you see this , it
09:18 means that the Group A Plus B is being multiplied
09:21 by the Group X Plus y . We could put
09:24 a time symbol between the groups , but we don't
09:26 have to because it's the default operation . In algebra
09:29 , the multiplication is just implied . So going back
09:33 to our problem two times five , if you wanted
09:36 to , you could just put each of the numbers
09:38 inside parentheses like this , and then you could get
09:41 rid of the multiplication site . Now , this can't
09:43 be confused with the number 25 . And since the
09:46 groups are right next to each other , you know
09:48 that you need to multiply them . Of course ,
09:51 it might seem strange to have just one thing inside
09:54 group symbols like this , but it's okay to do
09:56 that in math . An alternate way that you could
09:58 do . The same thing would be to put just
10:00 one of the numbers in parentheses like this again .
10:04 You won't confuse this with a two digit number ,
10:06 and you know that multiplication is implied . Okay ,
10:10 so we've learned that algebra is a lot like arithmetic
10:13 , but it involves unknown values or variables that we
10:16 need to solve for . And we learned that in
10:18 algebra , the multiplication sign is usually not shown because
10:22 it's the default operation . You can just assume that
10:25 two things right next to each other are being multiplied
10:28 . But why do we even care about algebra ?
10:30 Is it good for anything in the real world ?
10:33 Or is it just a bunch of tricky problems that
10:35 keep students busy in school ? Actually , algebra is
10:39 very useful for describing or modeling things in the real
10:42 world . It's a little hard to see that when
10:44 you're just looking at all these numbers and symbols on
10:47 the page of a math book , but it's a
10:49 lot easier to see when you start taking algebraic equations
10:52 and graphing their solutions . Graphing an equation is like
10:57 using its different solutions to draw simple lines and curves
11:00 that can be used to describe and predict things in
11:03 real life . For example , there's a whole class
11:06 of equations in algebra called linear equations because they form
11:10 straight lines when you graph them . Those sorts of
11:12 equations could help you describe the slope of the roof
11:16 or tell you how long it will take to get
11:17 somewhere . Another class of algebraic equations , called quadratic
11:22 equations , can be used to design telescope lenses or
11:26 describe how a ball flies through the air or predict
11:29 the growth of the population . So algebra is used
11:33 all the time in fields like science , engineering ,
11:36 economics and computer programming . And even though you might
11:39 not need algebra to get by in your day to
11:41 day life , so divide both sides by three .
11:45 That means X equals 42 . So in 321 yes
11:52 , all right now to see how much butter I
11:54 need , it's still a very useful part of math
11:58 . Thanks for watching math antics , and I'll see
12:00 you next time . Learn more at math antics dot
12:03 com
Summarizer

DESCRIPTION:

This video gives an overview of Algebra and introduces the concepts of unknown values and variables. It also explains that multiplication is implicit in Algebra.

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Algebra Basics: What Is Algebra? - Math Antics is a free educational video by Lumos Learning.

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