- THE ADMISSION FEE AT AN AMUSEMENT PARK IS $15.50
FOR CHILDREN AND $20.00 FOR ADULTS.
ON A CERTAIN DATE, 217 PEOPLE ENTERED THE PARK,
AND THE ADMISSION FEES COLLECTED TOTALED $3,782.
HOW MANY CHILDREN AND HOW MANY ADULTS WERE ADMITTED?
SO TO SOLVE THIS PROBLEM,
WE'RE GOING TO SET UP A SYSTEM OF EQUATIONS
AND THEN SOLVE THE SYSTEM.
WE'RE TRYING TO FIND HOW MANY CHILDREN AND HOW MANY ADULTS
ENTERED THE PARK ON THIS DAY, SO WE HAVE TWO UNKNOWN VALUES.
SO LET'S LET X = THE NUMBER OF CHILDREN,
AND WE'LL LET Y = THE NUMBER OF ADULTS.
NOW LET'S SORT THROUGH THE GIVEN INFORMATION.
THE FEE FOR EACH CHILD IS $15.50,
AND THE FEE FOR EVERY ADULT IS $20.00.
WE ALSO KNOW THERE WAS A TOTAL
OF 217 PEOPLE THAT ENTERED THE PARK ON THIS DAY,
AND THE TOTAL FEES COLLECTED WAS $3,782.
SO ONE EQUATION WILL REPRESENT THE AMOUNT OF MONEY COLLECTED,
AND THE OTHER EQUATION WILL BE THE NUMBER OF PEOPLE
THAT ENTERED THE PARK.
WELL, IF X IS THE NUMBER OF CHILDREN,
AND Y IS THE NUMBER OF ADULTS,
AND THERE WERE 217 PEOPLE IN THE PARK THAT DAY,
THIS TELLS US THAT X + Y = 217.
AND THEN AGAIN, EVERY CHILD HAS TO PAY $15.50,
AND EVERY ADULT HAS TO PAY $20.00,
RESULTING IN A TOTAL OF $3,782 WHICH MEANS 15.50
OR JUST 15.5 x X, THIS IS THE AMOUNT OF MONEY
COLLECTED FROM THE CHILDREN + $20.00 x Y,
THIS IS THE AMOUNT OF MONEY COLLECTED FROM THE ADULTS,
MUST EQUAL $3,782.00.
SO THIS IS OUR SYSTEM OF EQUATIONS.
IF WE WANT TO SOLVE THIS ALGEBRAICALLY,
WE CAN USE THE SUBSTITUTION METHOD
OR THE ELIMINATION METHOD.
SINCE WE CAN SOLVE THIS FIRST EQUATION HERE PRETTY EASILY
FOR X OR Y, I THINK WE'LL USE THE SUBSTITUTION METHOD.
SO IF WE SUBTRACT X ON BOTH SIDES OF THIS FIRST EQUATION,
WE CAN REWRITE THIS AS Y = 217 - X.
AND NOW WE'LL USE THIS EQUATION
AND PERFORM SUBSTITUTION INTO THE SECOND EQUATION.
SINCE Y = 217 - X, WE CAN SUBSTITUTE OR REPLACE
THIS Y HERE WITH 217 - X,
AND THIS WILL GIVE US AN EQUATION WITH ONE VARIABLE.
AND THEREFORE WE CAN SOLVE FOR X.
SO WE'D HAVE 15.5X + 20(217 - X) = 3,782.
SO NOW WE'LL CLEAR THE PARENTHESES AND SOLVE FOR X.
SO WE'LL HAVE 20 x 217.
THAT'S GOING TO BE 4,340 - 20X = 3,782.
LET'S COMBINE OUR LIKE TERMS HERE AND HERE.
15.5X - 20X IS GOING TO BE -4.5X.
NOW WE'LL SUBTRACT 4,340 ON BOTH SIDES.
THIS WOULD BE ZERO,
SO WE HAVE -4.5X = 3,782 - 4,340.
THAT'S GOING TO GIVE US -558.
LET'S GO AHEAD AND FINISH THIS UP HERE.
LET'S SOLVE THIS FOR X.
WE'LL DIVIDE BOTH SIDES BY -4.5.
THIS SIMPLIFIES TO 1X OR JUST X,
AND -558 DIVIDED BY -4.5 = 124.
REMEMBER, X REPRESENTS THE NUMBER OF CHILDREN,
SO WE STILL HAVE TO DETERMINE THE VALUE OF Y
WHICH WOULD BE THE NUMBER OF ADULTS.
SO WE CAN USE THIS EQUATION HERE,
X + Y = 217 TO DETERMINE Y,
OR WE CAN JUST USE THE EQUATION Y = 217 - X.
IF WE USE THIS EQUATION HERE,
WE KNOW THAT Y MUST BE EQUAL TO 217 - 124 = 93.
SO THIS TELLS US THAT THERE WERE 124 CHILDREN AND 93 ADULTS
ADMITTED TO THE PARK ON THIS GIVEN DAY.
AND REMEMBER, IF WE WERE TO GRAPH THESE TWO LINEAR EQUATIONS
ON THE SAME COORDINATE PLANE,
THE POINT OF INTERSECTION WOULD REPRESENT THE SAME SOLUTION.
SO LET'S GO AHEAD AND CHECK THAT.
SO HERE WE HAVE THE GRAPH OF THE TWO LINEAR EQUATIONS,
AND HERE IS OUR POINT OF INTERSECTION
WHERE THE X-COORDINATE = 124
WHICH REPRESENTS THE NUMBER OF CHILDREN ADMITTED,
AND THE Y-COORDINATE IS 93
WHICH REPRESENTS THE NUMBER OF ADULTS ADMITTED ON THAT DAY.
SO THIS DOES VERIFY OUR SOLUTION.