- WE WANT TO SOLVE THE GIVEN LINEAR EQUATION
IN ONE VARIABLE.
OUR GOAL IS TO ISOLATE THE VARIABLE
ON ONE SIDE OF THE EQUATION
AND WE DO HAVE SOME GUIDELINES HERE IN RED
THAT WE CAN REFER TO IN ORDER TO SOLVE
MULTISTEP LINEAR EQUATIONS.
THE FIRST STEP IN THIS EQUATION
IS GOING TO BE TO CLEAR THE PARENTHESIS.
SO I WILL HAVE TO DISTRIBUTE THE 8 HERE AND THE -2 HERE.
WE HAVE TO BE CAREFUL HERE WHEN DISTRIBUTING A NEGATIVE
AND WE ALSO HAVE SUBTRACTION.
LET'S GO AHEAD AND START ON THE LEFT.
WE'RE GOING TO HAVE 8 x X,
THAT'S 8X + 8 x 2 THAT'S 16
AND WE STILL HAVE 3 EQUALS
AND HERE WE WERE GOING TO HAVE - 2 x X, THAT'S -2X.
AND THEN BECAUSE OF THE SUBTRACTION
WE CAN THINK OF THIS AS A -4.
SO -2 x -4 WOULD BE 8.
SO WE ARE GOING TO WRITE 8.
TAKE A CLOSER LOOK AT THIS IN JUST A MOMENT
AND THEN WE'VE -11.
IT IS IMPORTANT THAT WE REALLY UNDERSTAND
WHAT'S HAPPENING HERE WHEN DISTRIBUTING A -2
ACROSS SUBTRACTION.
SO LET'S WRITE THIS OUT.
-2 x A QUANTITY X - 4.
WE WOULD REALLY HAVE -2 x X - -2 x 4.
SO HERE IS THE -2X AND THIS IS REALLY -8
AND SUBTRACTING A NEGATIVE IS SAME AS ADDING A POSITIVE.
SO THIS IS -2X + 8 BUT I THINK IT'S EASIER TO THINK THIS THING
AS -2 x -4, WHICH RESULTS IN 8.
SO WE WRITE DOWN 8.
OKAY, LETS GO AHEAD AND CONTINUE WITH OUR EQUATION NOW.
THE NEXT STEP IS GOING TO BE TO COMBINE THE LIKE TERMS
ON EACH SIDE OF THE EQUATION.
WE CAN COMBINE THESE TWO TERMS AS WELL AS THESE TWO TERMS.
SO WE HAVE 8X AND 16 + 3 IS 19.
HERE WE HAVE -2X AND HERE WE HAVE 8 - 11,
THAT'S GOING TO BE -3
AND INSTEAD OF WRITING + -3, WE WILL JUST WRITE - 3.
NOW NOTICE WE DO HAVE VARIABLE TERMS
ON BOTH SIDES OF THE EQUATION,
SO WE CAN ISOLATE X IF IT'S ON BOTH SIDES.
SO IF WE WANT TO UNDO THIS -2X
ON THE RIGHT SIDE OF THE EQUATION,
WE COULD ADD 2X TO BOTH SIDES.
AGAIN, THE REASON THIS WORKS - 2X + 2X WOULD BE 0,
SO NOW THE X TERM IS NO LONGER ON THE RIGHT SIDE.
SO NOW WHEN WE SIMPLIFY, WE HAVE 8X + 2X, THAT'S 10X + 19 = -3.
AND NOW WE HAVE OUR BASIC TWO STEP EQUATION.
WE WANT TO UNDO THIS ADDITION AND UNDO THIS MULTIPLICATION.
WE ARE GOING TO UNDO THE ADDITION FIRST,
SO WE WILL SUBTRACT 19 ON BOTH SIDES.
19 - 19 IS 0.
SO LET THE 10X = -3 - 19 WOULD BE -22
AND THEN THE LAST STEP IS TO DIVIDE BOTH SIDES BY 10.
SO 10 DIVIDED 10 IS 1.
SO WE HAVE 1X OR JUST X = -22/10ths
BUT THESE DO SHARE A COMMON FACTOR OF 2,
SO GO AHEAD AND WRITE THAT OUT.
WE CAN WRITE -22 AS -2 x 11 AND WE CAN WRITE 10 AS 2 x 5.
SO THERE IS THE COMMON FACTOR OF 2.
SO THIS SIMPLIFIES TO -11/5ths,
WHICH IS SIMPLIFIED IN FRACTION FORM.
THIS IS AN IMPROPER FRACTION.
SO IF WE WANT TO EXPRESS THIS AS A MIXED NUMBER,
WE WOULD HAVE TO DIVIDE 11 BY 5.
LET'S GO AHEAD AND DO THAT.
WE KNOW THAT MIXED NUMBER WOULD BE NEGATIVE,
SO WE ARE LEFT WITH 11 DIVIDED BY 5.
THERE ARE TWO 5s IN 11, 2 x 5 IS 10, THE REMAINDER OF 1.
SO -11/5ths IS EQUIVALENT TO -2
AND THE FRACTION PART IS A REMAINDER OVER THE DIVISOR,
SO WE HAVE 1/5th.
SO IT'S QUITE A BIT OF WORK
BUT THIS WOULD BE SOLUTION TO OUR EQUATION,
WHICH MEANS IF WE WENT BACK TO THE ORIGINAL EQUATION
AND SUBSTITUTED -11/5ths FOR X HERE AND HERE,
THE LEFT SIDE WOULD BE EQUAL TO THE RIGHT SIDE.
YOU MAY WANT TO VERIFY THAT FOR YOURSELF.
I HOPE THIS WAS HELPFUL.