- A MOVIE THEATER IS FILLED TO CAPACITY OF 350.
THE THEATER CHARGES $4.50 FOR CHILDREN, $7.50 FOR STUDENTS,
AND $12.50 FOR ADULTS.
THERE ARE HALF AS MANY ADULTS AS THERE ARE STUDENTS.
IF THE TOTAL TICKET SALES WAS $2,415;
HOW MANY CHILDREN, STUDENTS, AND ADULTS ATTENDED?
SO BECAUSE WE HAVE THREE UNKNOWN QUANTITIES,
WE'LL HAVE TO ASSIGN THREE VARIABLES FOR THESE VALUES.
SO LETS LET X EQUAL THE NUMBER OF CHILDREN,
LETS LET Y EQUAL THE NUMBER STUDENTS,
AND LETS LET Z EQUAL THE NUMBER OF ADULTS.
NOW LETS BEGIN TO WRITE OUR SYSTEM OF EQUATIONS.
WELL, SINCE THE THEATER IS FILLED TO CAPACITY,
X + Y + Z MUST EQUAL 350.
NEXT, WE'RE TOLD THE PRICE OF EACH TICKET
AND WE ALSO KNOW THE TOTAL SALES IS $2,415.
SO EVERY X IS WORTH 450, EVERY Y IS WORTH 750,
AND EVERY Z IS WORTH 1250.
SO WE CAN SAY THAT 4.5X, WHERE 4.5 REPRESENTS $4.50 + 7.5 x Y,
WHERE 7.5 REPRESENTS
$7.50 + 12.5 x Z MUST EQUAL THE TOTAL SALES OF $2,415.
AND WE'RE ALSO TOLD THAT THERE
ARE HALF AS MANY ADULTS AS THERE ARE STUDENTS.
SO IF THERE'S HALF AS MANY ADULTS AS THERE ARE STUDENTS,
THE NUMBER OF ADULTS, Z,
WOULD HAVE TO BE EQUAL TO THE NUMBER OF STUDENTS, Y,
DIVIDED BY TWO OR WE COULD DO ONE-HALF TIMES Y,
WHATEVER Y IS THE NUMBER OF STUDENTS,
IF WE DIVIDE BY TWO THAT WOULD GIVE US THE NUMBER OF ADULTS.
AND TO AVOID FRACTIONS HERE THOUGH,
WE COULD MULTIPLY BOTH SIDES OF THIS EQUATION BY 2 GIVING US
GIVING US THE EQUATION 2Z = Y OR Y = 2Z.
LET'S USE THIS EQUATION INSTEAD TO AVOID FRACTIONS.
SO OUR THIRD EQUATION IS Y = 2Z.
SO THIS IS THE SYSTEM OF EQUATIONS
THAT WE WANT TO SOLVE FOR X, Y, AND Z
TO DETERMINE HOW MANY CHILDREN, STUDENTS,
AND ADULTS WERE IN THE THEATER.
LET'S GO AHEAD AND DO THIS ON THE NEXT SLIDE.
TO SOLVE THIS SYSTEM, WE WANT TO FORM TWO EQUATIONS
WITH TWO UNKNOWNS AND THEN WE'LL SOLVE THAT SYSTEM
FOR ONE OF THE VARIABLES,
AND THEN PERFORM BACK SUBSTITUTION
TO FIND THE VALUE OF THE REMAINING VARIABLES.
AND SINCE THIS LAST EQUATION IS SOLVE FOR Y,
WE'RE GOING TO PERFORM SUBSTITUTION
FOR Y IN THE FIRST AND SECOND EQUATION.
SINCE Y = 2Z, WE CAN SUBSTITUTE 2Z FOR Y HERE AND 2Z FOR Y HERE.
THIS WILL GIVE US TWO EQUATIONS WITH JUST X AND Z.
SO PERFORMING THIS SUBSTITUTION INTO THE FIRST EQUATION,
WE WOULD HAVE X + 2Z + Z = 350.
COMBINING LIKE TERMS WE WOULD HAVE X + 3Z = 350.
THIS IS OUR FIRST EQUATION WITH 2 VARIABLES.
NOW I'LL PERFORM THE SAME SUBSTITUTION
INTO THE SECOND EQUATION.
SO THAT WOULD GIVE US 4.5X + 7.5 x 2Z + 12.5Z = 2,415.
WELL, 7.5 x 2Z = 15Z.
SO IF THIS IS 15Z + 12.5Z = 27.5Z.
SO NOW WE HAVE 4.5X + 27.5Z = 2,415.
SO THIS IS OUR SECOND EQUATION,
BUT, AGAIN, ONLY CONTAINS TWO VARIABLES;
THE VARIABLE X AND THE VARIABLE Z.
SO NOW WE'LL SOLVE EQUATION ONE
AND TWO AS A SYSTEM OF TWO EQUATIONS WITH TWO UNKNOWNS.
SO TO KEEP THIS ORGANIZED LET'S GO AHEAD
AND REWRITE EQUATION ONE AND EQUATION TWO.
WE HAVE X + 3Z = 350 AND THEN WE HAVE 4.5X + 27.5Z = 2,415.
AND NOW TO SOLVE THIS SYSTEM LETS GO AHEAD
AND SOLVE THIS FIRST EQUATION FOR X,
AND THEN PERFORM SUBSTITUTION INTO THE SECOND EQUATION.
SO IF WE SOLVE THIS EQUATION FOR X
WE'D HAVE TO SUBTRACT 3Z ON BOTH SIDES,
THAT WOULD GIVE US X = 350 - 3Z.
AND NOW WE'LL SUBSTITUTE 350 - 3Z FOR X
AND THE SECOND EQUATION.
AND THIS WILL FINALLY GIVE US ONE EQUATION WITH ONE UNKNOWN.
WE HAVE 4.5 x 350 - 3Z + 27.5Z = 2,415.
LET'S GO AHEAD AND FINISH SOLVING THIS ON THE NEXT SLIDE.
LET'S GO AHEAD AND DISTRIBUTE HERE,
4.5 x 350 = 1,575 AND THEN 4.5 x -3Z
WOULD BE -13.5Z + 27.5Z = 2,415.
COMBINING LIKE TERMS HERE,
-13.5Z + 27.5Z WOULD BE 14Z
AND I'LL SUBTRACT 1,575 FROM BOTH SIDES,
THAT'D GIVE US 14Z = 840 DIVIDE BOTH SIDES BY 14 AND WE HAVE Z,
WHICH IS THE NUMBER OF ADULTS, = 60.
SO WE GO BACK TO THE FIRST SLIDE,
LET'S GO AHEAD AND WRITE THIS DOWN.
IF Z = 60, WE KNOW THERE WERE 60 ADULTS.
NOW PERFORM BACK SUBSTITUTION TO FIND Y AND X.
WELL, WE KNOW THAT Y = 2 x Z.
SO WE HAVE Y = AGAIN, IF Z = 60,
WE HAVE 2 x 60 FOR Y, WHICH IS 120.
AND THEREFORE, THERE WOULD BE 120 STUDENTS.
NOTICE THERE ARE HALF AS MANY ADULTS AS THERE ARE STUDENTS
AS IT SAYS HERE.
AND FINALLY, TO FIND X NUMBER OF CHILDREN,
WE KNOW THAT X + Y + Z MUST = 350.
SO X, THE UNKNOWN, + Y, WHICH IS 120, + Z WHICH IS 60,
MUST = 350.
SO THIS WOULD BE X + 180 = 350.
SUBTRACTING 180 ON BOTH SIDES WE WOULD HAVE X = 170,
WHICH MEANS THERE ARE 170 CHILDREN IN THE THEATER.
I HOPE YOU FOUND THIS EXPLANATION HELPFUL.