00:06 hi i'm rob welcome to math antics in the
00:09 last two algebra videos we learned how
00:11 to solve simple equations that had only
00:13 one arithmetic operation in them but
00:15 often equations have many different
00:17 operations which make solving them a
00:20 little more complicated
00:21 in this video we're going to learn how
00:23 to solve equations that have just two
00:25 math operations in them
00:26 one addition or subtraction and one
00:28 multiplication or division
00:30 and the concepts you learn in this video
00:32 will help you solve even more
00:34 complicated equations in the future
00:36 now as you might expect equations that
00:39 have two arithmetic operations in them
00:41 are going to require two different steps
00:44 in other words to get the unknown all by
00:46 itself you'll need to undo two
00:50 but that doesn't sound too hard right i
00:51 mean we learned how to undo any
00:53 arithmetic operation in the last two
00:55 videos and that's true but there's a
00:57 couple reasons that make two-step
00:59 equations a little trickier to solve
01:02 the first is that there's a lot more
01:04 possible combinations of those two
01:05 operations and the second is that when
01:08 there's more than one operation you have
01:10 to decide what order to undo those
01:15 if you need to know what order to do
01:16 operations and just follow the order of
01:19 operations rules you did watch that
01:24 but the order of operations rules tells
01:27 us what order to do operations not what
01:29 order to undo them oh
01:32 then could we reverse the order since
01:35 we're undoing the operations now that's
01:39 well of course it is when solving
01:41 multi-step equations that's basically
01:44 what we're going to do
01:45 using the order of operations rules in
01:47 reverse can help us know what order to
01:52 but it can be a little tricky actually
01:53 putting it into practice
01:55 so to see how it works let's start by
01:58 solving a very simple two-step equation
02:01 two x plus two equals eight
02:04 in this equation the unknown value x is
02:07 involved in two different operations
02:09 addition and multiplication which is
02:11 implied between the first two and the x
02:14 and to undo those two operations we need
02:16 to use their inverse operations
02:18 subtraction and division
02:20 but the question is which one should we
02:22 do first like many things in life the
02:25 order we decide to do things in can make
02:30 oh come on there's gotta be an easier
02:33 first socks then shoes
02:36 fortunately in math we have a special
02:39 set of rules that tell us what order to
02:43 those rules tell us to do operations
02:45 inside parentheses or other groups first
02:48 and then we do exponents and then we do
02:50 multiplication and division and last of
02:52 all we do addition and subtraction
02:55 those are the rules you need to follow
02:57 when simplifying mathematical
02:58 expressions or equations but solving an
03:01 equation is different because we're
03:02 trying to undo any operations that the
03:05 unknown value is involved with so that
03:07 the unknown value will be all by itself
03:10 so when solving equations the best
03:12 strategy is to apply those order of
03:14 operations rules in reverse
03:16 using the reverse order of operations is
03:19 not the only way to solve a multi-step
03:21 equation but it's usually the easiest
03:23 way just like it's much easier to take
03:25 your shoes and socks off in the reverse
03:27 order that you put them on
03:32 are you sure it sucks before shoes
03:35 since the order of operations rules tell
03:37 us to do multiplication before we do
03:41 addition we should undo addition before
03:44 we undo multiplication
03:47 so first we undo the addition by
03:50 subtracting 2 from both sides of the
03:53 on the first side the plus 2 and the
03:55 minus 2 cancel each other out leaving
03:58 just 2x on that side
04:00 and on the other side we have eight
04:02 minus two which is six
04:05 next we can undo the multiplication by
04:08 dividing both sides of the equation by
04:11 on the first side the twos cancel
04:14 leaving x all by itself
04:16 and on the other side we have 6 divided
04:18 by 2 which is just 3.
04:21 there we've solved the equation using
04:23 the order of operations rules in reverse
04:26 and now we know that x equals three
04:29 that wasn't so bad was it let's try
04:32 solving another simple two-step equation
04:34 that has division and subtraction in it
04:36 x over two minus one equals four
04:40 again we're going to apply the order of
04:42 operations rules in reverse to undo the
04:45 subtraction and the division operations
04:47 since we would normally do the
04:49 subtraction last we're going to undo it
04:53 to undo the subtraction we add one to
04:55 both sides of the equation on the first
04:57 side the minus one and the plus one
05:00 cancel out leaving just x over two on
05:04 and on the other side we have four plus
05:08 and then to undo the divided by 2 we
05:11 need to multiply both sides by 2.
05:14 on the first side the 2s cancel leaving
05:16 x all by itself and on the other side we
05:19 have 2 times 5 which is 10. so our
05:22 answer is x equals 10.
05:25 those examples are pretty easy right but
05:27 solving two-step equations gets a bit
05:29 trickier thanks to a little something in
05:33 do you remember how parentheses are used
05:35 to group things in math and our order of
05:37 operations rules say that we're supposed
05:39 to do any operations that are inside
05:43 in other words we need to do operations
05:45 that are inside of groups first
05:47 well guess what that means that when
05:49 we're solving equations and undoing
05:51 operations we need to wait to do groups
05:56 to see what i mean let's solve this
05:58 equation which looks very similar to the
06:00 first one we solved
06:01 the only difference is that a set of
06:03 parentheses has been used to group this
06:07 and even though that might not seem like
06:08 much of a change makes a big difference
06:12 that's because in the original equation
06:14 this first two is only being multiplied
06:17 by the x but in the new equation it's
06:20 being multiplied by the entire quantity
06:24 and that's going to change how we solve
06:27 we're still going to follow our order of
06:29 operations rules in reverse but now that
06:31 the x plus 2 is inside parentheses which
06:34 means that it's part of a group we're
06:36 going to undo that operation last
06:39 since we're supposed to do operations in
06:41 groups first it means that we're going
06:42 to undo operations in groups last
06:45 so in this problem we should start by
06:48 undoing the multiplication that's
06:50 implied between the two and the group x
06:54 to do that we divide both sides of the
06:58 on the first side the two on the top and
07:01 the two on the bottom cancel leaving the
07:03 group x plus two on that side and on the
07:07 we have eight divided by two which is
07:11 that looks simpler already and we can
07:13 make it even simpler than that because
07:15 now that there's nothing else on that
07:17 side of the equal sign with the group x
07:20 we really don't even need the
07:21 parentheses anymore
07:23 next we just need to subtract two from
07:25 both sides on the first side the plus
07:28 two and the minus two cancel out leaving
07:32 and on the other side we have four minus
07:36 so for this equation x equals two
07:39 and now you can see how grouping
07:41 operations differently in our equation
07:43 results in different answers
07:45 let's try one more important example do
07:48 you remember the second equation we
07:50 solved x over two minus one equals four
07:54 in this equation the one is being
07:56 subtracted from the entire x over two
08:00 but take a look at this slightly
08:01 different equation this looks a lot like
08:03 the original equation but now that the
08:05 one is up on top of the fraction line
08:07 it's only being subtracted from the x
08:11 the x minus 1 on top forms a group
08:14 hold on how can x minus 1 be a group i
08:16 don't see any parentheses or brackets
08:19 ah that's a good question
08:21 in algebra the fraction line is used as
08:24 a way to automatically group things that
08:26 are above it or things that are below it
08:29 for example in this fancy algebraic
08:31 expression everything that's on top of
08:33 the fraction line forms a group and
08:35 everything on the bottom of the line
08:37 forms another group
08:38 of course we could put parentheses there
08:41 if we wanted to make it really clear but
08:44 grouping above and below a fraction line
08:47 is just implied in algebra
08:49 getting back to our new problem now that
08:52 we know that the x minus 1 on the top of
08:54 the fraction line is an implied group
08:56 as we learned in our last example we're
08:59 going to wait and undo the operation
09:01 inside that group last
09:03 so the first step is to undo the divided
09:06 by 2 by multiplying both sides of the
09:10 on the first side the 2 on the top and
09:12 the 2 on the bottom will cancel out
09:14 leaving just our implied group x minus 1
09:19 and on the other side
09:20 we have four times two which is eight
09:24 next we can undo the operation inside
09:26 the group by adding one to both sides on
09:29 the first side the minus one and the
09:31 plus one cancel leaving x all by itself
09:34 and on the other side we have eight plus
09:36 one which is nine so in this equation x
09:42 all right as you can see solving two
09:45 step equations is definitely more
09:47 complicated than single step equations
09:49 because there's so many different
09:51 combinations and different ways to group
09:54 but if you just take things one step at
09:56 a time and remember to undo operations
09:59 using the reverse order of operation
10:01 rules it will be much easier just pay
10:04 close attention to how things are
10:05 grouped in an equation and be on the
10:07 lookout for those implied groups on the
10:09 top and bottom of a fraction line
10:12 and because there are so many variations
10:14 of these two step equations it's really
10:17 important to practice by trying to solve
10:19 lots of different problems as always
10:21 thanks for watching math antics and i'll
10:25 learn more at mathantics.com
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