 ### Ex: Linear Equation Application (Cost of a Rental Car)

– A CAR RENTAL COMPANY
CHARGES \$210 PER WEEK PLUS \$.15 PER MILE
TO RENT A CAR. HOW MANY MILES CAN YOU DRIVE
IN ONE WEEK FOR \$336? SO WE WANT TO START
BY FOCUSING ON THIS FIRST SENTENCE. WE WANT TO BE ABLE
TO READ THIS AND THEN MOUNT AN EQUATION
TO MODEL THE TOTAL CHARGES FOR A RENTAL CAR. SO WE’LL START BY IDENTIFYING
THE VARIABLES THAT WE’LL USE
IN OUR EQUATION. THERE’S GOING TO BE
A TOTAL COST, A WEEKLY COST
AND A MILEAGE COST. SO LET’S LET BIG C EQUAL
THE TOTAL COST IN DOLLARS. LET’S LET W EQUAL THE NUMBER
OF WEEKS AND THEN WE’LL LET M EQUAL
THE NUMBER OF MILES DRIVEN. SO NOTICE THE TOTAL COST
IS AFFECTED BY THE NUMBER OF WEEKS
THAT YOU HAVE THE CAR AS WELL AS HOW MANY MILES
YOU DRIVE. SO THE EQUATION FOR THIS
SITUATION WOULD BE BIG C THE TOTAL COST IS GOING TO BE EQUAL TO \$210 x THE NUMBER
OF WEEKS SO THAT WOULD BE 210W + \$.15
PER MILE, WHICH WOULD BE 0.15 x M. AND NOW USING THIS EQUATION
HERE, WE SHOULD BE ABLE TO ANSWER
THE QUESTION. WE WANT TO KNOW HOW MANY MILES
YOU CAN DRIVE IN ONE WEEK FOR \$336. SO IF THE TIME IS ONE WEEK, WE
KNOW THAT W IS EQUAL TO ONE, THEN IF YOU HAVE A TOTAL
OF \$336 TO SPEND, THAT WOULD BE BIG C. SO NOW WE’LL SUBSTITUTE THESE
VALUES INTO THE EQUATION AND SOLVE FOR THE UNKNOWN
WHICH WILL BE M, THE NUMBER OF MILES. SO WE WOULD HAVE
336=210 x 1 + 0.15M. OF COURSE 210 x 1 IS JUST 210 AND THAT WILL SOLVE
THIS TWO STEP EQUATION FOR M SO WE’LL SUBTRACT 210
ON BOTH SIDES. THIS WOULD BE 0,
THIS WOULD BE 126=0.15M. SO NOW WE’LL DIVIDE BOTH SIDES
BY 0.15 TO SOLVE FOR M. SO THIS WOULD BE M, AND THEN
THIS QUOTIENT WILL TELL US HOW MANY MILES YOU CAN DRIVE. SO WE’D HAVE 126
DIVIDED BY 0.15 AND THEN WE’D HAVE TO MOVE
THE DECIMAL POINTS HERE BECAUSE REMEMBER WE WANT THIS
DIVISOR TO BE A WHOLE NUMBER SO I’LL MOVE THIS TWO
TO THE RIGHT AND THE SAME HERE AND SO I’LL MOVE
THE DECIMAL POINT UP AND THEN ADD TWO ZEROS,
ONE HERE AND ONE HERE. NOW WE WANT TO DETERMINE HOW
MANY 15’S THERE ARE IN 126, THAT WOULD BE 8, 8 x 5 IS 40, CARRY THE 4, 8 x 1 + 4
THAT WOULD BE 12. DIFFERENCE HERE IS 6,
WHICH IS LESS THAN 15, SO THE 8 IS CORRECT. BRING DOWN THE NEXT DIGIT. NOW WE’RE GOING TO KNOW
HOW MANY 15’S THERE ARE IN 60, THAT WOULD BE 4,
4 x 15 IS EXACTLY 60, DIFFERENCE IS 0, BUT WE STILL
HAVE ONE MORE DIGIT TO USE. BRING THE 0 DOWN HERE
AND THERE ARE 0 15’S IN 0 SO WE PUT A 0 HERE,
0 x 15 IS 0, AND WE SUBTRACT THE DIFFERENCE
OF 0. WE’VE USED ALL THE DIGITS SO
WE KNOW THAT M IS EQUAL TO 840 WHICH MEANS YOU WOULD BE ABLE
TO DRIVE A TOTAL OF 840 MILES.