[THEME MUSIC]
WHETHER WE CALL THEM WORD
PROBLEMS IN ALGEBRA
OR CALL THEM REAL
APPLICATIONS OF ALGEBRA,
SOME PEOPLE CALL
THEM SIMPLY IMPOSSIBLE.
BUT THAT'S NOT TRUE.
IF YOU CAN REMEMBER
THE ALPHABET,
YOU CAN REMEMBER HOW TO
APPROACH AND OVERCOME
A WORD PROBLEM IN ALGEBRA.
IT'S AS SIMPLE AS
P, Q, R, S, T.
WE PREVIEW.
WE IDENTIFY THE QUESTION.
WE REWRITE OR TRANSLATE.
SOLVE.
AND THEN TEST
THE SOLUTION.
SO, LET'S TRY ONE OF THOSE
"IMPOSSIBLE" WORD PROBLEMS.
HMMM, TWO CONSECUTIVE
INTEGERS WHICH ADD UP TO 35.
LET'S SEE -- P Q R S T
YOU HAVE TO KNOW
WHAT THE PROBLEM SAYS,
KNOW WHAT YOU'RE
LOOKING FOR.
TRANSLATE THE ENGLISH
WORDS INTO AN EQUATION --
THE LANGUAGE OF ALGEBRA.
SOLVE THE EQUATION AND
CHECK YOUR ANSWER.
P Q R S T
IS JUST A WAY OF HELPING
YOU TO REMEMBER THAT.
LET'S TAKE IT STEP BY STEP.
P - FOR PREVIEW.
READ THE WHOLE PROBLEM. UNDERSTAND WHAT IT SAYS.
SEE IF THERE ARE ANY WORDS
YOU DON'T UNDERSTAND.
HOW ABOUT THE WORD INTEGER?
AN INTEGER MEANS A WHOLE
NUMBER, LIKE 1 OR 2, 15, 63.
NOT A FRACTION OR DECIMAL.
AND WHAT ABOUT TWO
CONSECUTIVE INTEGERS?
THAT'S JUST TWO WHOLE
NUMBERS IN A ROW,
LIKE 3 - 4 OR 7 - 8.
NEXT STEP --
Q FOR QUESTION.
WHAT ARE YOU LOOKING FOR, AND
CAN WE NAME IT WITH A LETTER?
USUALLY WE CAN, AND
THAT'S IMPORTANT
BECAUSE WE NEED A WAY TO
REPRESENT OUR UNKNOWN --
WE CALL IT THE VARIABLE --
TO MAKE AN EQUATION.
SINCE OUR WORD PROBLEM ASKS
FOR CONSECUTIVE INTEGERS HERE,
WE CAN NAME
ONE INTEGER "X"
AND THE OTHER, SINCE
THEY'RE CONSECUTIVE,
HAS TO BE X PLUS 1.
STEP 3: R - REWRITE.
THAT IS, TRANSLATE
THE WORD PROBLEM
INTO THE LANGUAGE
OF ALGEBRA.
WE WRITE AN EQUATION.
2 CONSECUTIVE WHOLE NUMBERS
WHICH ADD UP TO 35.
THIS IS JUST A PLAIN
OLD LINEAR EQUATION.
AND YOU KNOW HOW TO
SOLVE LINEAR EQUATIONS.
LET'S WORK THROUGH IT.
FIRST, YOU COMBINE
LIKE TERMS.
THEN SUBTRACT 1
FROM BOTH SIDES
TO GET THE 2X BY ITSELF.
THEN DIVIDE BOTH
SIDES BY 2
TO GET FROM 2X DOWN
TO A SIMPLE X.
AND THERE'S YOUR SOLUTION!
OR IS IT?
WE SOLVED FOR X,
BUT WE HAVEN'T QUITE ANSWERED OUR ORIGINAL QUESTION.
AT THIS POINT YOU
ALWAYS WANT TO GO BACK
AND ASK YOURSELF WHAT WAS THE
ORIGINAL QUESTION EXACTLY?
IN THIS CASE WE WANTED
TWO WHOLE NUMBERS.
WE NAMED ONE X AND
THE OTHER X + 1.
SO IF X IS 17, THEN THE
OTHER NUMBER MUST BE 18.
WE'LL FIND OUT IF THIS IS
CORRECT IN THE FINAL STEP,
T FOR TEST
WHERE YOU CHECK YOUR ANSWER
TO MAKE SURE IT WORKS.
THE ANSWER WORKS BECAUSE
WE FOLLOWED OUR ALPHABET
P Q R S T
PREVIEW, QUESTION,
REWRITE, SOLVE, AND TEST.
READY TO HANDLE
ONE BY YOURSELF?
TRY THIS WORD PROBLEM.
HERE'S A HINT:
NOTICE THAT WE'RE TALKING
NOW ABOUT EVEN INTEGERS.
THINK ABOUT HOW FAR APART
EVEN INTEGERS HAVE TO BE.
PAUSE THE PROGRAM NOW.
GIVE IT A TRY.
AND WHEN YOU ARE READY,
CLICK PLAY AND WE WILL
COMPARE SOLUTIONS.
WE TACKLE THIS WORD PROBLEM
USING OUR ALPHABET:
P Q R S T.
PREVIEW: TWO NUMBERS ADDED
EQUALS A MULTIPLE
OF A THIRD NUMBER.
THE PROBLEM IS EASY
ENOUGH TO UNDERSTAND,
AND I UNDERSTAND CONSECUTIVE,
EVEN, AND INTEGER.
SO I MOVE ON TO NOTICE
EXACTLY WHAT IS ASKED:
THE QUESTION
FIND 3 NUMBERS.
LET'S CALL THE
SMALLEST ONE X.
THEY'RE EVEN NUMBERS, SO THE
NEXT ONE WILL BE LARGER BY 2
LIKE 8, 10, 12, OR 14.
SO WE CALL THE
SECOND ONE X + 2.
AND THE LARGEST OF
THE THREE IS X + 4.
CAN WE REWRITE IT
INTO AN EQUATION?
WE CAN.
THE LARGEST PLUS THE SMALLEST
EQUALS SIX TIMES THE MIDDLE.
ONCE WE'VE EXPRESSED OUR
PROBLEM AS AN EQUATION,
IT'S NO BIG JOB TO FIND THE
RIGHT ANSWER -- STEP BY STEP.
ADD LIKE TERMS
ON THE LEFT.
APPLY THE DISTRIBUTIVE
LAW TO THE RIGHT.
SIMPLIFY BY SUBTRACTING
4 FROM BOTH SIDES
AND SUBTRACTING
6X FROM BOTH SIDES.
AND THEN WE GET
DOWN TO A SIMPLE X
BY DIVIDING BOTH
SIDES BY NEGATIVE 4,
WHICH TELLS US THAT
X EQUALS NEGATIVE 2.
IS THAT THE ANSWER?
NO, IT'S THE BEGINNING
OF THE ANSWER.
BECAUSE THE QUESTION WAS
FIND 3 CONSECUTIVE
EVEN INTEGERS.
X, OR NEGATIVE 2,
IS THE FIRST ONE.
X + 2, OR ZERO,
IS THE SECOND.
AND X + 4, OR POSITIVE 2,
IS THE THIRD.
ARE THEY THE CORRECT THREE
CONSECUTIVE EVEN INTEGERS?
WE CAN EASILY TEST TO SEE
IF THEY FIT THE STORY.
MINUS 2 PLUS 2 EQUALS ZERO,
AND 6 TIMES ZERO IS ZERO.
ZERO EQUALS ZERO.
OUR SOLUTION WORKS!
WORD PROBLEMS MAY
SEEM IMPOSSIBLE,
BUT THEY CAN BE SOLVED
IF WE GET THEM TRANSLATED
INTO THE RIGHT EQUATION.
TRY ANOTHER ONE.
LOOK IMPOSSIBLE?
DON'T PANIC.
FIRST, DO YOU UNDERSTAND
ALL THE WORDS HERE?
GOOD. PREVIEW DONE.
LET'S JUMP STRAIGHT
TO THE QUESTION
FIND SARAH'S AGE.
FIRST, WE NEED TO
NAME THE VARIABLE.
BUT IT PRESENTS ME
WITH A LITTLE PROBLEM,
BECAUSE THERE ARE TWO
RULES I LIKE TO FOLLOW
IN NAMING A VARIABLE.
AND IN THIS CASE,
THE TWO RULES DISAGREE.
RULE 1) USE X FOR THE QUANTITY
YOU ARE TRYING TO FIND.
THAT WOULD BE SARAH'S
AGE IN THIS CASE.
RULE 2) WHERE YOU HAVE
SEVERAL UNKNOWNS,
USE X FOR THE SMALLEST ONE.
AND THAT WOULD
BE JENNY'S AGE.
HERE I'M GOING TO OPT
FOR THE SECOND RULE.
YOU WILL SEE LATER WHY
IT WORKS BETTER THIS WAY.
WE WILL CALL
JENNY'S AGE "X".
SO, SARAH'S IS 2X.
AND WE'LL HAVE TO
REMEMBER THAT
WHEN WE GET OUR
ANSWER FOR X.
NOW LET'S TRANSLATE THE
WORD PROBLEM TO ALGEBRA:
SINCE JENNY'S AGE - X -
PLUS SARAH'S AGE - 2 X -
EQUALS 69,
WE HAVE OUR EQUATION AND
WE'RE READY TO GO TO WORK.
WE SEE THAT 3X IS 69,
THEN 1X IS 69 DIVDED BY 3
OR 23.
AND WE HAVE AN ANSWER.
WELL, ALMOST AN ANSWER.
REMEMBER, I DECIDED TO
LET JENNY'S AGE BE X.
SO JENNY'S AGE IS 23.
BUT OUR QUESTION WAS:
FIND SARAH'S AGE.
SARAH IS TWICE AS
OLD AS JENNY.
SO SARAH IS 46, AND
THAT'S OUR REAL ANSWER.
WE TEST OUR ANSWER BY ADDING
OUR VALUES FOR X AND 2X --
23 AND 46.
THE SUM IS 69 AND OUR
ANSWER CHECKS OUT.
NOW, WHY DIDN'T I USE SARAH'S
AGE AS X IN THIS PROBLEM?
AFTER ALL, IT IS WHAT
WE'RE LOOKING FOR.
THE ANSWER IS
I LIKE TO AVOID FRACTIONS IN
MY EQUATIONS WHENEVER I CAN.
BY TAKING THE SMALLEST
UNKNOWN -- JENNY'S AGE -- AS X,
WE GET THIS
SIMPLE EQUATION.
BUT IF I TOOK
SARAH'S AGE AS X,
THEN JENNY'S AGE WOULD
BE HALF AS MUCH, OR ½X,
AND MY EQUATION WOULD
HAVE A FRACTION IN IT.
WOULD THIS EQUATION WORK?
YES. IT DOES TRANLATE
THE WORD PROBLEM CORRECTLY
AND IT WOULD GIVE YOU SARAH'S
AGE DIRECTLY AS YOUR ANSWER.
BUT IT WOULD TAKE YOU A
LITTLE LONGER TO GET THERE
BECAUSE OF THAT FRACTION.
FOLLOW ALONG AS I SHOW YOU
THE EXTRA STEPS INVOLVED.
AND DON'T FORGET, WHEN
YOU GET YOUR ANSWER
GO BACK AND ASK YOURSELF
WHAT WAS THE QUESTION?
AND WHAT WAS X?
THE QUESTION WAS:
WHAT IS SARAH'S AGE?
AND THIS TIME WE LET
SARAH'S AGE BE X,
SO WE HAVE OUR
ANSWER DIRECTLY.
YOU'LL FIND MORE EXAMPLES
OF WORD PROBLEMS LIKE THESE
IN YOUR TEXTBOOK.
HERE'S A PROBLEM FOR
SOME IMMEDIATE PRACTICE.
DON'T LET THE
WORDS SCARE YOU.
LOOK FOR THE QUESTION.
DO IT THE SAME WAY
WE DID THE LAST PROBLEM.
PAUSE THE PROGRAM UNTIL
YOU HAVE AN ANSWER.
CLICK PLAY, AND WE
WILL COMPARE NOTES
ON HOW WE HANDLED
THIS WORD PROBLEM.
I'LL ASSUME WE ALL UNDERSTAND
ABOUT RENT AND UTILITIES.
SO LET'S SKIP THE PREVIEW
AND SEE WHAT THE QUESTION IS.
NOTICE THAT THE QUESTION
IS FREQUENTLY WRITTEN
AS A STATEMENT.
BUT THE MEANING
IS A QUESTION.
"WHAT IS YOUR MONTHLY
UTILITY BILL?"
AND IN THIS CASE,
I HAVE NO TROUBLE
DECIDING WHAT TO CALL X.
THE MOST IMPORTANT UNKNOWN
IS THE UTILITY BILL.
THE SMALLEST UNKNOWN
IS THE UTILITY BILL.
SO WHAT WILL X BE?
(YOU'RE WAY AHEAD OF ME)
THE UTILITY BILL.
NOW I NEED TO WRITE
A TERM FOR THE RENT.
RENT IS TEN TIMES
THE UTILITY BILL,
SO I USE 10 TIMES "X".
AND IN THE REWRITE
OR TRANSLATION STEP,
I TURN ALL OF IT
INTO AN EQUATION --
A SIMPLE EQUATION THAT WON'T
TAKE LONG AT ALL TO SOLVE.
JUST REMEMBER TO USE DIVISION
TO GET RID OF THE 11
TO GET A VALUE
OF 95 FOR "X".
ARE WE DONE? OR IS
THERE SOMETHING ELSE
YOU SHOULD BE ASKING
YOURSELF AT THIS POINT?
YOU HAVE X.
IS IT THE ANSWER?
WELL, WHAT WAS
THE QUESTION?
AND WHAT WAS X?
THE QUESTION WAS WHAT IS
YOUR MONTHLY UTILITY BILL?
WHAT DID WE WRITE AS X?
THAT SAME UTILITY BILL.
SO YOUR FINAL
ANSWER IS 95.
CHECK YOUR ANSWER
BY SUBSTITUTING 95 FOR X
IN THE EQUATION.
AND IT WORKS.
NOTICE THAT WE COULD HAVE
DONE THE PROBLEM THE SAME WAY
EVEN IF WE'D BEEN
ASKED "WHAT IS THE RENT?"
INSTEAD OF "WHAT
IS THE UTILITY BILL?"
WE'D JUST ADD A FINAL STEP,
BECAUSE THE RENT IS TEN
TIMES THE UTILITY BILL
OR $950.
AND THAT'S JUST ONE
MORE REMINDER TO ASK:
WHAT WAS I LOOKING FOR
IN THE FIRST PLACE?
AND WHAT WAS IT
THAT I NAMED X?
BY NOW, P Q R S T
AND THE EXACT MEANING
OF EACH LETTER,
SHOULD BE AS FAMILIAR TO YOU AS THE NAME OF YOUR BEST FRIEND.
IF NOT, STUDY THEM WELL!
WRITE THEM DOWN. THINK
ABOUT THEM. AND USE THEM.
EVERY PROBLEM IS DIFFERENT,
BUT THESE STEPS WILL SAVE
YOU LOTS OF TROUBLE.
BE SURE TO TRY SOME PRACTICE
PROBLEMS FROM YOUR TEXTBOOK.
HOW MANY PROBLEMS?
AS MANY AS IT TAKES TO
BE A "P Q R S T" WIZARD.
[THEME MUSIC]
CAPTIONS PROVIDED BY
THE DISABILITY INSTRUCTIONAL SUPPORT CENTER