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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