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- IN ORDER TO SOLVE THE GIVEN LINEAR EQUATION IN ONE VARIABLE,
OUR GOAL IS TO ISOLATE THE VARIABLE
ON ONE SIDE OF THE EQUATION.
SO WE CAN SEE HERE BELOW IN RED,
WE HAVE SOME GUIDELINES TO HELP US SOLVE THIS TYPE OF EQUATION.
THE FIRST THING YOU'LL PROBABLY NOTICE ABOUT THIS EQUATION
IS THERE'S FRACTIONS INVOLVED.
AND THE FIRST STEP IS GOING TO BE TO CLEAR
THE FRACTIONS FROM THE EQUATION.
TO DO THIS, WE'RE GOING TO MULTIPLY
BOTH SIDES OF THE EQUATION
BY WHAT WOULD BE THE LEAST COMMON DENOMINATOR OF 6 AND 4,
WHICH IS THE SAME AS THE LEAST COMMON MULTIPLE OF 6 AND 4,
WHICH WOULD BE 12.
SO WE'RE GOING TO MULTIPLY BOTH SIDES OF THE EQUATION
OR EACH TERM BY 12.
NOW, IF YOU HAVE A HARD TIME
DETERMINING THE LEAST COMMON MULTIPLE OF 6 AND 4,
WE COULD JUST MULTIPLY THESE TWO DENOMINATORS TOGETHER,
WHICH WOULD BE 24, AND MULTIPLY EVERYTHING BY 24,
BUT THAT WOULD REQUIRE ADDITIONAL SIMPLIFYING LATER.
SO WE'RE GOING TO GO AHEAD AND MULTIPLY EVERYTHING HERE
BY 12 ON BOTH SIDES OF THE EQUATION.
AND WHEN MULTIPLYING BY A FRACTION
I'M GOING TO GO AHEAD AND PUT THE 12/1.
AND NOW WHEN YOU DETERMINE THESE PRODUCTS
IT SHOULD CLEAR THE FRACTIONS.
LOOKING AT THIS FIRST PRODUCT,
NOTICE HOW 6 AND 12 HAVE A COMMON FACTOR OF 6.
THERE'S ONE 6 IN 6 AND TWO 6s IN 12.
SO WE'RE LEFT WITH 2 x THE QUANTITY X - 1 - 12 x 6
THAT'S 72 EQUALS AND HERE WE HAVE A COMMON FACTOR OF 4
BETWEEN 12 AND 4.
THERE'S ONE 4 IN 4 AND THREE 4s IN 12.
SO WE'RE LEFT WITH 3 x THE QUANTITY X - 3.
SO NOTICE NOW WE NO LONGER HAVE FRACTIONS.
SO THE NEXT STEP IS GOING TO BE TO CLEAR THE PARENTHESIS,
SO WE'LL DISTRIBUTE HERE AND HERE, AS WELL AS HERE AND HERE.
SO WE'RE GOING TO HAVE 2X - 2 - 72 EQUALS THIS
WILL BE 3X - 9.
NOW LET'S GO AHEAD AND COMBINE THE LIKE TERMS
ON BOTH SIDES OF THE EQUATION.
WE CAN COMBINE THESE TWO TERMS, SO WE'D HAVE 2X.
AND WE'RE GOING TO THINK OF THIS AS -2 - 72
THAT'S GOING TO BE -74 OR - 74.
NOW, REMEMBER OUR GOAL IS TO ISOLATE THE VARIABLE,
AND WE CAN'T DO THIS WHEN WE HAVE X TERMS
ON BOTH SIDES OF THE EQUATION.
SO TO GET THE VARIABLE TERMS ON ONE SIDE,
WE EITHER HAVE TO SUBTRACT 2X ON BOTH SIDES
OR SUBTRACT 3X ON BOTH SIDES.
AND TO KEEP THE X TERM POSITIVE,
I'M GOING TO GO AHEAD AND SUBTRACT 2X ON BOTH SIDES.
HERE WE HAVE 2X - 2X THAT'S 0, THAT'S WHY WE DID THAT.
AND WE HAVE -74 = 3X - 2X WOULD BE X, AND THEN - 9.
LET'S GO AHEAD AND FINISH THIS UP HERE.
WE'RE GOING TO HAVE -74 = X - 9.
SO TO ISOLATE X ON THE RIGHT SIDE OF THE EQUATION,
WE WANT TO UNDO THIS -9.
WELL, THE OPPOSITE OF -9 WOULD BE 9.
SO WE'LL ADD 9 TO BOTH SIDES OF THE EQUATION.
NOTICE HERE WE HAVE -9 + 9 THAT WOULD BE 0.
SO ON THE LEFT SIDE WE HAVE -74 + 9
AND THAT WOULD GIVE US -65 EQUALS
AND ON THE RIGHT SIDE WE JUST HAVE X.
SO -65 = X, WHICH IS EQUIVALENT TO X = -65,
WHICH MEANS IF WE REPLACE X WITH -65 IN THE ORIGINAL EQUATION,
IT WOULD SATISFY THIS EQUATION OR MAKE THIS EQUATION TRUE.
WE'RE NOT GOING TO TAKE THE TIME TO CHECK THAT NOW,
BUT YOU MAY WANT TO JUST TO VERIFY THIS SOLUTION.
OKAY, HOPE YOU FOUND THIS HELPFUL.