[THEME MUSIC]
WHAT CAN WE DO WITH
EXPRESSIONS LIKE THESE?
WHAT CAN WE DO?
CAN WE GO AHEAD AND ADD
WHEREVER IT SAYS ADD?
SUBTRACT WHEREVER
IT SAYS SUBTRACT?
WHEN CAN WE?
WHEN CAN'T WE?
WE DON'T KNOW YET, BUT
WE'RE ABOUT TO FIND OUT.
[ELEPHANT TRUMPETING]
WHAT IF YOU LOOKED OUT
YOUR WINDOW ONE MORNING
AND SAW 3 ELEPHANTS
IN YOUR YARD?
AND TWO MORE IN YOUR
NEIGHBOR'S YARD?
YOU MIGHT SAY, "WHOA!
5 ELEPHANTS!"
LOOK WHAT YOU'VE DONE.
YOU'VE COUNTED THE OCCURRENCES
OF ONE KIND OF WIDGET -
AN ELEPHANT.
3 OCCURRENCES PLUS 2
OCCURRENCES MAKE 5 -
ALL OCCURRENCES OF THE SAME
KIND OF THING: ELEPHANTS.
IN ALGEBRA IT'S THE SAME.
BUT WE DEAL WITH
MATHEMATICAL WIDGETS
AND WE NAME THEM
WITH LETTERS.
THEN WE WRITE THEM IN A
PHRASE CALLED AN EXPRESSION,
TO DESCRIBE WHAT WE WANT
TO DO WITH OUR WIDGETS.
IN THIS EXPRESSION, X
IS ONE KIND OF WIDGET.
WE COUNT 3 OF X AND THEN 2
MORE OF X AND WE HAVE 5 X.
X'S ARE JUST LIKE ELEPHANTS.
AND NOTICE THIS: THE
ANSWER WASN'T JUST 5.
NO, WE HAD 5
ELEPHANTS OR 5 X'S
EVEN THOUGH ALL WE DID
WAS ADD THE NUMBERS.
THAT'S BECAUSE
IN EACH CASE
WE ARE DEALING WITH
HOW MANY WIDGETS WE HAVE.
AND THAT GIVES
US A BASIC RULE:
IN ALGEBRA, THE
PARTS OF EXPRESSIONS
THAT YOU'RE ADDING OR SUBTRACTING ARE CALLED TERMS.
WHENEVER YOU'RE ADDING JUST
ONE KIND OF TERM - OR WIDGET -
WE CALL THEM "LIKE TERMS".
WHENEVER YOU'RE
ADDING LIKE TERMS,
JUST ADD THE NUMBERS AND
BRING THE LETTERS ALONG.
ALWAYS? ALWAYS.
BECAUSE WHAT CHANGES IS HOW
MANY OF THEM THERE ARE.
THERE WERE 6 OF THEM,
THEN THERE WERE THREE MORE.
SO NOW THERE ARE ALTOGETHER
9 OCCURRENCES OF X.
THE NUMBERS WITHOUT THE LETTERS
ARE WHAT WE CALL COEFFICIENTS.
WE SIMPLY ADD THE COEFFICIENTS WHENEVER WE HAVE LIKE TERMS -
ALL z OR ALL y OR ALL A.
ADD THE COEFFICIENTS AND BRING
THE LETTERS ALONG FOR THE RIDE.
SUBTRACTION? SAME THING.
IN FACT THIS KIND OF TERM - A
COEFFICIENT TIMES A LETTER -
WILL FOLLOW ALL THE
RULES YOU LEARNED
FOR WORKING WITH SIGNED
NUMBERS IN ARITHMETIC.
HERE'S A LINE FOR
COUNTING. REMEMBER?
EVERY DOT IS ONE
STEP, ONE NUMBER.
EVERYTHING YOU ADD - EVERYTHING
PLUS, EVERYTHING POSITIVE -
IS A STEP TO THE RIGHT.
ADDING POSITIVE 2
LOOKS LIKE THIS:
COUNT OFF 2 STEPS
TO THE RIGHT.
HERE ARE 3 EXAMPLES
OF ADDITION.
IN ALL 3, WE'RE
ADDING A POSITIVE 2
AND WE TAKE 2 STEPS
TO THE RIGHT.
BUT WHAT IF WE'RE ADDING
NEGATIVE NUMBERS INSTEAD?
WHAT WOULD YOU EXPECT?
JUST THE OPPOSITE.
AND THAT'S JUST WHAT YOU GET.
ADDING A NEGATIVE NUMBER -
HERE IT'S A NEGATIVE 3 -
TAKES YOU BACK, NEGATIVE,
TO THE LEFT.
YOU MIGHT THINK THAT
ADDING A NEGATIVE NUMBER
IS JUST THE SAME AS
SUBTRACTING A POSITIVE NUMBER.
AND IT IS.
WE CAN WRITE THESE SAME 3
EXAMPLES THAT WAY IF WE WANT,
WITHOUT CHANGING A THING.
WE STILL MOVE 3 STEPS BACK
TO THE LEFT IN EVERY CASE.
THAT'S A GOOD
RULE TO REMEMBER.
IN FACT, YOU'D BE
HOPELESSLY LOST WITHOUT IT.
ONCE AGAIN: ADD A
NEGATIVE NUMBER,
SUBTRACT A POSITIVE NUMBER,
SAME THING - NO
DIFFERENCE AT ALL!
2 PLUS NEGATIVE 3 IS
THE SAME THING AS 2 - 3.
X PLUS NEGATIVE Y IS
THE SAME THING AS X - Y.
HANG ONTO THAT ONE!
AND THE FINAL CASE - THIS
IS TOUGH FOR SOME PEOPLE,
BECAUSE WE DON'T LIKE
TWO NEGATIVES AT ONCE -
TO SUBTRACT A NEGATIVE
NUMBER, YOU ADD.
MINUS AND
MINUS MAKES PLUS.
NEGATIVE AND NEGATIVE
MAKES POSITIVE.
ALWAYS?
IN ADDITION AND
SUBTRACTION, ALWAYS.
NO MATTER WHAT KIND OF TERM
YOU'RE ADDING OR SUBTRACTING.
WHEREVER YOU SEE TWO
MINUS SIGNS TOGETHER,
REPLACE THEM WITH A PLUS.
IT'LL BE MUCH
EASIER TO DEAL WITH.
IF WE PUT ALL 4 OF
THOSE RULES TOGETHER,
WE END UP WITH ONLY TWO:
GO RIGHT OR GO LEFT.
GO RIGHT, OR POSITIVE,
TO ADD POSITIVE NUMBERS OR
SUBTRACT NEGATIVE NUMBERS.
GO LEFT, OR NEGATIVE,
TO ADD NEGATIVE NUMBERS OR SUBTRACT POSITIVE NUMBERS.
SO WHAT DOES ALL THIS
HAVE TO DO WITH ELEPHANTS,
WIDGETS AND LIKE TERMS?
IT TELLS YOU EXACTLY HOW TO
HANDLE A PROBLEM LIKE THIS -
AN ALGEBRA EXPRESSION
WITH SEVERAL TERMS.
THEY ARE ALL LIKE TERMS,
ALL W's.
SO ALL YOU HAVE TO HANDLE
ARE THE COEFFICIENTS,
AND YOU ALREADY
KNOW HOW TO DO THAT.
SO FIRST YOU HANDLE IT AS
A TOTALLY NUMERIC PROBLEM,
ALL COEFFICIENTS.
THEN MAKE SURE THAT YOU'VE KEPT
THE "W" WHEREVER IT BELONGS,
AND ESPECIALLY IN YOUR ANSWER.
THE RIGHT ANSWER IS 6W,
NOT JUST 6.
BUT YOU GET THE ANSWER BY
WORKING WITH ONLY THE NUMBERS.
"W" JUST GOES ALONG
FOR THE RIDE,
BECAUSE EVERY TERM
OF THE EXPRESSION
IS TALKING ABOUT "W'S".
SO, READY TO TRY
ONE ON YOUR OWN?
PAUSE NOW, WHILE YOU
SIMPLIFY THIS EXPRESSION.
WATCH YOUR SIGNS CAREFULLY.
START PLAY AGAIN, AS SOON AS
YOU'VE WRITTEN YOUR ANSWER.
WHEN YOU'RE ADDING
OR SUBTRACTING,
THE FIRST QUESTION TO
ALWAYS ASK - ALWAYS:
DO WE HAVE LIKE TERMS,
THE SAME KIND OF WIDGET
EVERYWHERE IN THE PROBLEM,
OR ARE WE TRYING TO MIX APPLES
AND MANGOS AND ORANGES?
BECAUSE THAT WON'T WORK.
SO, ARE THEY
ALL LIKE TERMS?
YES THEY ARE - ALL X.
SO THEN WE JUST THINK THROUGH
IT AS A PROBLEM IN COEFFICIENTS
WITH THEIR PLUS AND MINUS SIGNS.
BUT DON'T FORGET TO TAKE
THE X'S ALONG ALL THE WAY.
NOTICE VERY CLEARLY THAT
THE ANSWER IS NOT 2.
THE ANSWER IS 2X.
BECAUSE WHAT WE'RE REALLY
DOING HERE IS COUNTING X'S,
JUST AS WE STARTED BY COUNTING
THE ELEPHANTS IN YOUR BACKYARD.
AND SPEAKING OF ELEPHANTS,
WHAT DO YOU THINK
WOULD HAPPEN
IF WE TRIED TO ADD 3
ELEPHANTS AND 2 TIGERS?
THAT'S DEFINITELY
NOT GOING TO WORK.
ELEPHANTS AND
TIGERS ARE UNLIKE;
THEY ARE DIFFERENT,
DISSIMILAR, DISTINCT.
AND THEY DON'T
GET ALONG WELL -
NOT THE SAME KIND
OF WIDGET AT ALL.
IN ALGEBRA WE CALL SUCH
THINGS UNLIKE TERMS,
AND WE TREAT THEM MUCH MORE
CAREFULLY THAN LIKE TERMS.
WE CAN'T JUST THROW 3X AND 2Y
TOGETHER AND SIMPLIFY THINGS.
X AND Y ARE UNLIKE
TERMS AND THEY DON'T MIX.
SAME FOR X TO THE POWER
2 AND X TO THE POWER 3.
X SQUARED AND X CUBED
ARE UNLIKE TERMS
AND WE CAN'T PUT
THEM TOGETHER.
BUT WHAT IF SOME TERMS ARE
ALIKE AND SOME ARE UNLIKE.
WHAT DO WE DO?
CAN WE SIMPLIFY THE
EXPRESSION AT ALL?
WHAT DO YOU THINK?
THINK "YES" AND "NO":
NO, WE CAN'T PUT THE
UNLIKE TERMS TOGETHER;
AND YES, WE CAN
SIMPLIFY SOME
BY PUTTING THE LIKE
TERMS TOGETHER.
"A" AND NEGATIVE 8A
ARE LIKE TERMS,
SO WE COMBINE THEM
INTO NEGATIVE 7A.
"2B" AND "5B" ARE
LIKE TERMS,
AND WE COMBINE THEM
INTO POSITIVE 7B.
3C IS UNLIKE ANY OTHER TERM,
SO IT DOESN'T CHANGE AT ALL.
THE RESULT?
NEGATIVE 7A,
PLUS 7B, MINUS 3C.
THAT'S AS SIMPLE AS
WE CAN MAKE IT.
HERE'S ANOTHER ONE
FOR YOU TO TRY.
SAME IDEA. WHAT WOULD
YOU DO WITH THIS ONE?
GO AHEAD, PAUSE THE
SHOW AND GIVE IT A TRY.
SIMPLIFY THE LIKE TERMS.
LEAVE THE UNLIKE TERMS ALONE.
WRITE DOWN YOUR ANSWER, AND
THEN LET'S WORK IT TOGETHER.
GOT IT? WE START BY
LOOKING FOR LIKE TERMS.
AND WE FIND THEM: 2X
AND 7X, 3Y AND Y.
WE WORK OUT THE SIGNS
2X - 7X
-3Y + Y
AND END UP WITH
-5X - 2Y - 5Z
OR PERHAPS YOU GOT SOMETHING
A LITTLE DIFFERENT,
THE SAME TERMS BUT
IN DIFFERENT ORDER.
GOOD NEWS! YOU'RE
STILL RIGHT,
NO MATTER WHICH WAY
THEY'RE ARRANGED,
JUST AS LONG AS EACH
TERM IS CORRECT.
REMEMBER THE COMMUTATIVE
LAW FROM ARITHMETIC?
IT SAYS THAT IN SOME CASES
THE ORDER OR SEQUENCE
DOESN'T MATTER.
IT'S TRUE FOR ADDITION.
WE CAN ADD TWO NUMBERS
IN EITHER ORDER
AND WE STILL GET
THE SAME ANSWER.
MULTIPLICATION, TOO.
THE ORDER DOESN'T CHANGE
THE ANSWER AT ALL.
BOTH OF THESE COMMUTATIVE LAWS
ARE GOING TO BE VERY USEFUL
WHENEVER YOU GO HUNTING
FOR LIKE TERMS.
WHAT DOES THAT TELL
YOU ABOUT THIS PROBLEM?
DO YOU SEE ANY LIKE TERMS TO
SIMPLIFY IN THIS EXPRESSION?
SURE YOU DO. LOOK AT
THE FIRST PARENTHESES.
MULITIPLY, YOU GET 2XY.
OR HOW ABOUT 2YX? THEY'RE
THE SAME THING, REMEMBER?
YOU MAKE IT 2XY.
WHY? BECAUSE YOU SEE
A 3XY COMING UP NEXT.
THEN YOU MULTIPLY
NEGATIVE Y BY 4X.
YOU COULD WRITE IT NEGATIVE 4YX,
BUT NEGATIVE 4XY IS BETTER.
YOU MAKE THEM LOOK
LIKE LIKE TERMS,
BECAUSE THEY ARE
LIKE TERMS.
2XY + 3XY - 4XY
FROM THERE IT'S ALL
ABOUT COEFFICIENTS,
AND WE SOLVE IT ALL THE
WAY DOWN TO A SIMPLE XY.
GOT IT?
NOW WE'RE READY TO DO PROBLEMS
THAT COMBINE ADDITION,
SUBTRACTION, MULTIPLICATION
AND DIVISION.
TO BE SURE YOU CAN HANDLE
THE MULTIPLICATION,
YOU MAY WANT TO REVIEW
THE MODULE ON EXPONENTS.
IT DOES GET A
LITTLE COMPLICATED
WHEN YOU PUT ALL FOUR
OPERATIONS TOGETHER AT ONCE -
ADDITION, SUBTRACTION,
MULTIPLICATION (INCLUDING EXPONENTS) AND DIVISION.
TO HANDLE THAT, YOU MAY FEEL
LIKE YOU NEED A NEW BRAIN.
BUT ALL YOU REALLY
NEED IS A RECIPE:
STEP-BY-STEP DIRECTIONS TO
TELL YOU WHERE TO BEGIN
AND WHAT TO DO NEXT,
AND NEXT, AND NEXT -
UNTIL YOU'RE DONE.
FOLLOW THIS LIST. FIRST YOU
HANDLE BRACKETS OR PARENTHESES,
THEN EXPONENTS,
MULTIPLICATION OR DIVISION,
AND ADDITION OR SUBTRACTION.
FINALLY, CLEAN IT ALL UP
LEFT TO RIGHT.
LET'S TRY AN EXAMPLE.
WHICH CAME FIRST IN OUR RECIPE,
MULTIPLICATION OR ADDITION?
MULTIPLICATION DID.
SO WE GET 2 TIMES 3 IS 6,
PLUS 4 IS 10.
BUT NOTICE HOW THE ORDER CHANGES
WHEN WE BRACKET THE 3 + 4?
NOW WE NEED TO USE
THE RULE FOR BRACKETS.
AND WHEN IN OUR RECIPE LIST
DO WE DEAL WITH BRACKETS?
FIRST. MULTIPLICATION
COMES LATER.
WHICH MEANS THAT THE CORRECT
ANSWER TO THIS PROBLEM
IS FIRST, 3 PLUS 4 IS 7,
AND THEN THE MULTIPLICATION,
2 TIMES 7 IS 14 -
A VERY DIFFERENT ANSWER
BECAUSE WE HAD TO FOLLOW
A DIFFERENT ORDER.
NOW, BACK TO THAT COMPLICATED
PROBLEM WE SAW BEFORE.
LOOKS SCARY, DOESN'T IT?
BUT WE'RE GOING TO DO
IT BY USING OUR RECIPE.
WHAT COMES FIRST?
BRACKETS.
I SEE 3Y TIMES X AT THE END.
I COULD WRITE IT
3YX, BUT I DON'T.
I MAKE IT 3XY TO MATCH THE
5XY I SEE AT THE BEGINNING.
WHAT'S NEXT?
EXPONENTS.
WE HAVE AN X CUBED
AND AN X SQUARED,
BUT WE CAN'T SIMPLIFY
EITHER OF THEM YET.
NEXT.
MULTIPLICATION?
THERE STILL ISN'T
ANYTHING TO SIMPLIFY.
DIVISION?
THE WHOLE MIDDLE TERM IS
A PROBLEM IN DIVISION,
SO LET'S FOCUS ON
THAT MIDDLE TERM.
REMEMBER, YOU CAN HANDLE
COEFFICIENTS - THE NUMBERS -
SEPERATELY.
DIVIDING NUMBERS
IS PRETTY EASY.
HANDLE THE POWERS
OF X SEPARATELY, TOO.
REMEMBER YOUR
EXPONENT RULES: TO DIVIDE,
YOU SUBTRACT THE POWERS -
TOP MINUS BOTTOM.
AND THE Y IS JUST Y;
YOU LEAVE IT ALONE.
SO YOU THINK OF IT IN
PIECES, LIKE THIS:
2 "POINT" 4 DIVIDED BY
"POINT" 6 EQUALS 4.
X CUBED DIVIDED BY X SQUARED
- SUBTRACT 2 FROM 3 -
EQUALS X TO THE
POWER 1, OR JUST "X"
AND Y = Y.
THEN PUT ALL THE PIECES BACK
TOGETHER AGAIN, LIKE THIS.
SO WE PUT OUR SIMPLIFIED MIDDLE
TERM BACK IN ITS POSITION,
AND WE SUDDENLY FIND OURSELVES
WITH THREE LIKE TERMS.
WE FINISH UP BY ADDITION OR
SUBTRACTION FROM LEFT TO RIGHT,
WHICH IS AS SIMPLE AS
5 MINUS 4 PLUS 3,
TAKING THE XY
ALONG FOR THE RIDE.
AND THAT'S IT -
THE WHOLE STORY ABOUT ADDITION
AND SUBTRACTION IN ALGEBRA.
MAKE SURE YOU REALLY
UNDERSTAND IT NOW,
BECAUSE YOU'RE
GOING TO NEED IT
FOR ALL KINDS OF
THINGS LATER ON.
IT'S NOT HARD.
YOU'LL ALWAYS BE
IN GOOD SHAPE
IF YOU JUST REMEMBER
TO DO THINGS IN ORDER,
FOLLOW THE RECIPE, AND ONLY
ADD OR SUBTRACT LIKE TERMS.
IF YOU MIX THE TIGERS
WITH THE ELEPHANTS,
YOU'RE IN DEEP TROUBLE.
LET'S TRY
ANOTHER EXAMPLE.
IT'S GOOD TO PRACTICE
WHEN THE STEPS ARE STILL
FRESH IN YOUR MIND.
I'LL GIVE YOU A HELPFUL HINT.
YOU HAVE ONLY TWO STEPS HERE:
FIRST DIVISION, AND THEN
ADDITION AND SUBTRACTION.
SO PAUSE THE SHOW NOW,
GET YOUR ANSWER,
AND THEN COME BACK
AND WE'LL LOOK AT IT.
GOT IT?
WE'RE LOOKING AT TWO
DIVISION PROBLEMS.
THE FIRST IS SO SIMPLE YOU
CAN DO IT IN YOUR HEAD.
TO DIVIDE, SUBTRACT
THE EXPONENTS.
'A' TO THE THIRD POWER
DIVIDED BY 'A' TO THE FIRST
GIVES US 'A' SQUARED.
AND OUR FIRST TERM
IS DONE - IT'S 3A²B.
IN THE SECOND TERM, WE HAVE SOME
COEFFICIENTS TO WORK ON FIRST.
12 DIVIDED BY 2 IS 6.
THE 'A' SQUARED IS ALONE
AND DOESN'T CHANGE.
B TO THE FIFTH DIVIDED BY
B TO THE FOURTH (5 - 4)
IS B TO THE FIRST, OR B,
WHICH GIVES US OUR
SECOND TERM: 6A²B
AND SUDDENLY WE
FIND OURSELVES
WITH TWO LIKE TERMS TO BE
ADDED - THEY'RE BOTH "A²B"'S
AND THERE ARE THREE
PLUS SIX OF THEM,
FOR A FINAL ANSWER
OF 9A²B MINUS C.
WE SOLVED THIS PROBLEM BY
SIMPLIFYING EACH TERM IN ORDER
AND THEN LOOKING
FOR LIKE TERMS.
TRY SOME MORE EXAMPLES
FROM YOUR TEXTBOOK.
PRACTICE HELPS YOUR
NEW KNOWLEDGE
REALLY STICK IN YOUR BRAIN.
BUT IF ANYTHING STILL
SEEMS A LITTLE FUZZY,
YOU MAY WANT TO READ
MORE ABOUT IT FIRST
OR REVIEW PART
OF THIS PROGRAM.
EITHER WAY, PRACTICE UNTIL
SIMPLIFICATION IS EASY.
[THEME MUSIC]
CAPTIONS PROVIDED BY
THE DISABILITY INSTRUCTIONAL
SUPPORT CENTER
AT MISSION COLLEGE