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- IN ORDER TO SOLVE THE GIVEN LINEAR EQUATION IN ONE VARIABLE
OUR ULTIMATE GOAL IS TO ISOLATE THE VARIABLE
ON ONE SIDE OF THE EQUATION.
BUT THE FIRST THING YOU'LL PROBABLY NOTICE
ABOUT THIS EQUATION IS THAT IT CONTAINS DECIMALS.
LOOKING AT OUR GUIDELINES
FOR SOLVING MULTI-STEP LINEAR EQUATIONS
THE FIRST STEP IS TO CLEAR ANY FRACTIONS OR DECIMALS.
SO FOR SOLVING THIS EQUATION BY HAND
IT WILL BE HELPFUL TO ELIMINATE OR CLEAR THE DECIMALS
BY MULTIPLYING BY A POWER OF 10
BASED UPON THE NUMBER OF DECIMAL PLACES.
REMEMBER THE SHORTCUT FOR MULTIPLYING BY 10
IS TO MOVE THE DECIMAL POINT TO THE RIGHT ONE PLACE.
THE SHORTCUT FOR MULTIPLYING BY 100
IS TO MOVE THE DECIMAL POINT TWO PLACES RIGHT AND SO ON.
SINCE WE HAVE ONE DECIMAL PLACE HERE, ONE DECIMAL PLACE HERE,
BUT TWO DECIMAL PLACES HERE
IF WE WANT TO CLEAR THE DECIMALS
WE'LL HAVE TO MULTIPLY EVERY TERM BY 100.
SO LET'S GO AHEAD AND DO THAT.
SO AGAIN, MULTIPLYING BY 100 DOES HAVE THE EFFECT
OF MOVING THE DECIMAL POINT TO THE RIGHT TWO PLACES
SO FOR THIS FIRST PRODUCT 100 x 2X = 200X.
100 x 18.2,
IF WE MOVE THE DECIMAL POINT TO THE RIGHT TWO PLACES
WE DO HAVE TO ADD A 0 AND WE WOULD HAVE 1,820.
MOVING THE DECIMAL POINT TO THE RIGHT TWICE
HERE WE WOULD HAVE 280X AND HERE WE WOULD HAVE 148.
SO BECAUSE THIS ONE DECIMAL HAD TWO DECIMAL PLACES
WE HAD TO MULTIPLY EVERYTHING BY 100.
NOW LOOKING AT THE EQUIVALENT EQUATION
NOTICE HOW WE DO HAVE X TERMS ON BOTH SIDES OF THE EQUATION.
IF OUR GOAL IS TO ISOLATE X
WE DO HAVE TO HAVE IT ON ONE SIDE OF THE EQUATION.
SO TO GET THE VARIABLE TERMS ON ONE SIDE,
WE EITHER HAVE TO SUBTRACT 200X ON BOTH SIDES
TO ELIMINATE THIS VARIABLE TERM
OR WE HAVE TO SUBTRACT 280X ON BOTH SIDES
TO ELIMINATE THIS TERM.
NOTICE IF WE SUBTRACT 200X ON BOTH SIDES
WE WOULD HAVE A +X TERM ON THE RIGHT SIDE
AND IF WE SUBTRACTED 280X ON BOTH SIDES,
WE WOULD HAVE A -X TERM ON THE LEFT SIDE.
IT DOESN'T MATTER WHICH METHOD WE USE
BUT SINCE MOST PEOPLE PREFER TO HAVE THE VARIABLE TERM
ON THE LEFT SIDE
I'M GOING TO GO AHEAD AND SUBTRACT 280X ON BOTH SIDES.
NOTICE HERE WE DO HAVE 0 NOW.
AND NOW LET'S GO AHEAD AND SIMPLIFY HERE.
WE'RE GOING TO HAVE -80X + 1,820 = 148.
AND NOW, WE HAVE OUR BASIC TWO-STEP EQUATION.
WE WANT TO UNDO THIS MULTIPLICATION HERE
AS WELL AS THIS ADDITION HERE.
SO WE'LL UNDO THE ADDITION FIRST.
WE'LL SUBTRACT 1,820 ON BOTH SIDES.
THIS WOULD BE 0 SO WE HAVE -80X EQUALS THIS DIFFERENCE HERE.
IF WE'RE DOING THIS BY HAND
REMEMBER SUBTRACTING 1,820 IS EQUIVALENT TO ADDING -1,820.
SO THIS IS EQUIVALENT TO 148 + -1,820.
SO WHEN ADDING INTEGERS WITH DIFFERENT SIGNS
WE ACTUALLY SUBTRACT THEIR ABSOLUTE VALUES
AND KEEP THE SIGN OF THE NUMBER WITH THE LARGER ABSOLUTE VALUE.
ANOTHER WAY TO THINK OF THIS IS WE CAN SEE THE NEGATIVE
IS GOING TO OUTWEIGH THE POSITIVE.
SO THIS SUM WHICH IS THE SAME AS THIS DIFFERENCE
IS GOING TO BE NEGATIVE
AND NOW WE'LL JUST SUBTRACT THEIR ABSOLUTE VALUES
AND WE ALWAYS SUBTRACT THE SMALLER ABSOLUTE VALUE
FROM THE LARGER ABSOLUTE VALUE.
SO WE'LL HAVE +1,820
AS THE ABSOLUTE VALUE OF THIS NEGATIVE NUMBER
AND THEN WE'LL SUBTRACT THE ABSOLUTE VALUE OF 148
WHICH IS 148.
SO HERE WE HAVE TO BORROW.
10 - 8 = 2, BORROW, WE HAVE 11 - 4 = 7
AND HERE WE HAVE 7 - 1 = 6 AND 1.
BUT REMEMBER WE ALREADY DETERMINED
THAT THIS SUM WHICH IS THE SAME AS THIS DIFFERENCE
IS GOING TO BE NEGATIVE.
SO WE HAVE -1,672
AND THEN SINCE -80X MEANS -80 x X
WE'LL DIVIDE BOTH SIDES BY -80.
HERE WE HAVE 1X OR JUST X.
AND NOW WE NEED TO DETERMINE THIS QUOTIENT.
WE KNOW A NEGATIVE DIVIDED BY A NEGATIVE IS POSITIVE
AND SINCE THE EQUATION CONTAINED DECIMALS,
WE'RE GOING TO GO AHEAD AND GIVE THE SOLUTION AS A DECIMAL.
SO WE'LL PERFORM THIS DIVISION. WE'LL HAVE 1,672 DIVIDED BY 80.
SO WE NEED TO FIGURE OUT HOW MANY 80s THERE ARE IN 167
THAT'S GOING TO BE 2. 2 x 80 = 160.
HERE WE HAVE A DIFFERENCE OF 7, BRING DOWN THE 2,
THERE ARE ZERO 80s IN 70.
SO WE PUT A 0 HERE, 0 x 80 = 0. SUBTRACT, THAT'S 72,
AND NOW WE NEED TO ADD A DECIMAL.
DECIMAL POINT WOULD BE HERE
WHICH WE'LL MOVE INTO THE QUOTIENT
AND NOW WE CAN ADD A DIGIT OF 0 IN THE TENTHS PLACE VALUE
AND BRING IT DOWN.
AND THIS IS GOOD NEWS BECAUSE THERE ARE NINE 80s IN 720.
SO PUT A 9 HERE, 9 x 80 = 720
AND WE SUBTRACT AND THE DIFFERENCE IS 0.
SO THIS DECIMAL TERMINATES AND OUR SOLUTION IS 20.9.
REMEMBER WE KNOW IT'S POSITIVE
BECAUSE WE HAVE A NEGATIVE DIVIDED BY A NEGATIVE.
THAT WILL DO IT FOR THIS EXAMPLE.
I HOPE YOU FOUND THIS HELPFUL.