- WELCOME TO THE VIDEO
ON SOLVING EQUATIONS USING MULTIPLE STEPS. THERE WAS A PREVIOUS VIDEO THAT DISCUSSED HOW TO SOLVE
ONE-STEP EQUATIONS. LET'S GO AHEAD AND TAKE A LOOK AT HOW WE'RE GOING
TO APPROACH THIS. HERE ARE THE STEPS WE'LL FOLLOW
TO SOLVE EQUATIONS. STEP ONE, WE'LL USE
THE MULTIPLICATION PRINCIPLE TO CLEAR ANY FRACTIONS
OR DECIMALS IF THEY ARE PRESENT. STEP TWO, WE'LL MOVE
ANY PARENTHESES BY USING THE DISTRIBUTIVE
PROPERTY, AGAIN, IF THERE ARE PARENTHESES. STEP THREE, WE WILL ADD
AND SUBTRACT AS NEEDED TO GET ALL VARIABLE TERMS
ON ONE SIDE AND ALL CONSTANT TERMS
ON THE OTHER. STEP FOUR, WE WILL MULTIPLY OR
DIVIDE TO SOLVE WITH A VARIABLE. AND THEN STEP FIVE,
WE'LL CHECK OUR SOLUTIONS. SO LET'S GO AHEAD
AND GIVE IT A TRY. NUMBER ONE HERE, THERE'S NO
FRACTIONS OR DECIMALS, STEP TWO,
THERE'S NO PARENTHESIS, STEP THREE, WE WILL ISOLATE
THE VARIABLE TERMS. WHAT WE'LL DO HERE
IS SUBTRACT 7 ON BOTH SIDES THAT WILL LEAVE US WITH 3X = 36. AND THE LAST STEP, WE NEED
TO UNDO THIS MULTIPLICATION, SO WE'LL DIVIDE BOTH SIDES BY 3.
AND WE HAVE OUR SOLUTION X = 12. SO TO CHECK THIS 3 x 12
WOULD BE 36 + 7 = 43. AND IT CHECKS. MOVING ON TO NUMBER TWO. AGAIN THERE'S NO DECIMALS
OR FRACTIONS, NO PARENTHESES, SO THE FIRST THING WE DO
IS TO ISOLATE THE VARIABLE TERM, WE ADD OR SUBTRACT. THIS IS A +4,
SO TO UNDO THE +4, WE WILL SUBTRACT 4
ON BOTH SIDES, THIS WILL LEAVE US WITH -2X
OR -2X = 6, SINCE THAT SIMPLIFIES TO 0. AND THE LAST STEP,
SINCE WE ARE SOLVING FOR X, WE'LL DIVIDE BY A -2. NOW, AGAIN,
THE REASON WE'RE DIVIDING HERE IS THE -2 IS ATTACHED
TO THE X BY MULTIPLICATION. SO NOW WE SIMPLIFY, THIS WILL GIVE US X
AND 6 DIVIDED BY -2 = -3. FOR THE SAKE OF TIME
WE WILL NOT CHECK THIS ONE. MOVING ON TO NUMBER THREE,
WE DO HAVE A FRACTION INVOLVED, IT IS OPTIONAL
BUT WE WILL CLEAR THE FRACTION. AND THE WAY WE DO THAT IS WE'RE
GOING TO MULTIPLY EVERY TERM BY WHAT WOULD BE
THE COMMON DENOMINATOR, IN THIS CASE IT SHOULD BE 4. SO LET'S FIRST REWRITE THIS. SO WE'RE GOING TO MULTIPLY BOTH
SIDES OF THE EQUATION BY 4 SO YOU CAN THINK OF DISTRIBUTING
ON THE LEFT SIDE, NOTICE THE FOURS HERE
WOULD CANCEL. WE HAVE -X - 36 = -20. SO THE IDEA HERE IS THIS
EQUATION IS EASIER TO SOLVE WITHOUT THE FRACTION INVOLVED. SO NEXT STEP,
WE'LL ADD 36 TO BOTH SIDES. THIS WOULD GIVE US -X
ON THE LEFT SIDE AND THE RIGHT SIDE
WE'D HAVE +16. SO IF WE DIVIDE BOTH SIDES
BY -1, WE WOULD HAVE X = -16. OKAY, ON NUMBER FOUR
WE HAVE DECIMALS INVOLVED. WELL, IN ORDER TO ELIMINATE
THE DECIMALS WHAT WE CAN DO IS
MULTIPLY EVERYTHING BY 10. REMEMBER IF YOU MULTIPLY
EVERYTHING BY 10 YOU MOVE THE DECIMAL
TO THE RIGHT ONE PLACE. SO THE RESULT WOULD BE
AN EQUIVALENT EQUATION AND THAT EQUATION WOULD BE
43 = 15X - 50. AND THEN WE FOLLOW
THE SAME STEPS, WE WILL ISOLATE THE VARIABLE
FIRST BY ADDING OR SUBTRACTING. IN THIS CASE WE'LL ADD 50, THIS WOULD GIVE US 93
ON THE LEFT = 15X ON THE RIGHT, DIVIDE BOTH SIDES BY 15 BECAUSE THIS IS ATTACHED
BY MULTIPLICATION. SIMPLIFYING,
WE'D HAVE X ON THE RIGHT AND ON THE LEFT WE HAVE 93/15. NOW THIS DOES SIMPLIFY. 93 IS THE SAME AS 3 x 31
AND 15 IS THE SAME AS 3 x 5. SO --
THIS IS SIMPLIFIED TO 31/5. NOW, SINCE THE ORIGINAL EQUATION
WAS IN DECIMAL FORM WE MAY WANT TO WRITE THIS
AS 6.2. THOSE ANSWERS ARE EQUIVALENT BUT AGAIN, SINCE THE EQUATION
WAS IN DECIMAL FORM WE MAY WANT TO LEAVE OUR ANSWER
IN DECIMAL FORM. LET'S GO AHEAD AND TAKE A LOOK
AT A FEW MORE. NOW, BEFORE WE START
TO SOLVE THIS WE CAN SEE THAT THERE ARE
SOME LIKE TERMS ON THE SAME SIDE
OF THE EQUAL SIGN. SO LET'S TAKE A LOOK
AT THE LEFT SIDE FIRST. 8X - 10X WOULD GIVE US
A -2X + 7. AND ON THE RIGHT WE HAVE
TWO CONSTANT TERMS, AND 10 + 12 WOULD GIVE US 22. SO ON THE RIGHT SIDE
WE HAVE 22 + 3X. NOW WHAT'S DIFFERENT
ABOUT THIS EQUATION IS WE HAVE THE VARIABLE TERM
ON BOTH SIDES OF THE EQUATION. WELL, IF OUR GOAL IS TO ISOLATE
THE VARIABLE, OF COURSE, IT NEEDS TO BE
ON ONE SIDE. SO WE EITHER NEED TO SUBTRACT 3X
ON BOTH SIDES OR ADD 2X ON BOTH SIDES DEPENDING ON WHAT SIDE
YOU WANT THE VARIABLE TO BE ON. I'M GOING TO GO AHEAD
AND ADD 2X TO BOTH SIDES. SO ON THE LEFT SIDE
NOW WE HAVE 7 ON THE RIGHT SIDE WE HAVE
22 + 5X. SO SINCE X IS ON THE RIGHT SIDE
WE KNOW WHEN WE FINISH WE WANT TO HAVE SOMETHING IN THE
FORM OF SOME NUMBER EQUALS X. SO THE NEXT STEP
IS TO ISOLATE THE VARIABLE BY ADDING OR SUBTRACTING. SO WE'LL UNDO THIS +22
BY SUBTRACTING 22 ON BOTH SIDES. THIS WOULD GIVE US -15
IS EQUAL TO--THAT'S 0, 5X. LET'S GO AHEAD AND REWRITE THIS
OVER HERE. THE NEXT STEP WOULD BE
TO ISOLATE THE VARIABLE BY MULTIPLYING OR DIVIDING. IT'S ATTACHED BY MULTIPLICATION
SO WE NEED TO DIVIDE BY +5 RESULT WILL BE X OR 1X
ON THE RIGHT SIDE, ON THE LEFT SIDE WE HAVE -3.
SO OUR SOLUTION IS X = -3. NOW ON NUMBER SIX, THE FIRST THING WE'LL DO
IS CLEAR THE FRACTIONS. SO WHAT WE CAN DO, THERE'S
A COUPLE WAYS TO APPROACH THIS, BUT THE MOST OBVIOUS ONE
MIGHT BE TO MULTIPLY BOTH SIDES
OF THE EQUATION BY 3. AND NOTICE HOW WHAT HAPPENS HERE
IS THIS 3 SIMPLIFIES WITH THIS 3 SO WE'RE LEFT WITH
2 x 6X + 1 = 18. NOW I COULD'VE MULTIPLIED
BY THE RECIPROCAL 3/2 BUT I'LL GO AHEAD
AND LEAVE IT IN THIS FORM. NOW WE'LL CLEAR THE PARENTHESES
SO WE HAVE 12X + 2 = 18. AND AGAIN, THE NEXT STEP IS WE
UNDO THE ADDITION OR SUBTRACTION SO WE'LL SUBTRACT 2
ON BOTH SIDES AND I'LL WRITE THE RESULT
UP HERE. WE'D HAVE 12X ON THE LEFT,
THAT'S 0 AND WE'D HAVE 16 ON THE RIGHT. DIVIDING BY 12 ON BOTH SIDES-- AGAIN, WE'RE DIVIDING BECAUSE THAT 12 IS ATTACHED
TO THE X BY MULTIPLICATION. SO WE HAVE X = 16/12,
BUT THIS DOES SIMPLIFY AND WE CAN WRITE 16 AS 4 x 4
AND 12 AS 4 x 3, SO WE'D HAVE 4/3 OR 1 1/3
AS OUR SOLUTION. LET'S GO AHEAD AND TAKE A LOOK
AT ONE MORE. THE FIRST THING WE NOTICE
THERE'S NO DECIMALS OR FRACTIONS SO THE NEXT STEP WOULD BE TO CLEAR THE PARENTHESES
BY DISTRIBUTING. NEXT STEP IS WE'LL COMBINE
OUR LIKE TERMS. AGAIN, WE HAVE X TERMS
ON BOTH SIDES, SO WE NEED TO GET THE X TERMS
ON ONE SIDE SO WE EITHER NEED TO SUBTRACT 5X
ON BOTH SIDES OR SUBTRACT 3X ON BOTH SIDES. I WILL GO AHEAD AND SUBTRACT 5X
ON BOTH SIDES, SO WE HAVE TO DIVIDE BY -2
TO ISOLATE THE VARIABLE. THIS WOULD GIVE US 1X OR X AND ON THE LEFT WE'D HAVE
24 DIVIDED BY -2 = -12. I WOULD LIKE TO PAUSE NOW
AND GO BACK TO THE FIRST EXAMPLE AND SHOW YOU HOW YOU CAN CHECK
THIS ON THE GRAPHING CALCULATOR. ON NUMBER ONE,
WE SAID THE SOLUTION WAS X = 12. SO WHAT WE CAN DO IS ASSIGN 12
THE VALUE OF X ON THE GRAPHING CALCULATOR. SO WE TYPE IN "12, STORE X,"
ENTER. NOW WHAT WE CAN DO IS TYPE IN THE LEFT SIDE
OF THE EQUATION 3X + 7 AND SEE IF WE GET 43. AND WE CAN SEE THAT CHECKS. LET'S GO AHEAD
AND TRY NUMBER TWO AS WELL. WE'LL STORE -3 IN FOR X,
"-3, STORE X," ENTER. AND THEN WE'LL TYPE IN
THE LEFT SIDE AGAIN, "4 - 2X," AND SEE IF WE GET 10. THIS IS A VERY NICE WAY
TO CHECK YOUR WORK AS YOU SOLVE THESE EQUATIONS.