- THIS VIDEO WILL ADDRESS HOW TO SOLVE EQUATIONS GRAPHICALLY.
WE'LL TALK ABOUT THE INTERSECTION METHOD
AND THE ZERO METHOD.
REMEMBER THAT A SOLUTION TO AN EQUATION IS A VALUE OR VALUES
THAT MAKE THE EQUATION TRUE.
WE MUST OFTEN SOLVE EQUATIONS ALGEBRAICALLY,
BUT IN THIS VIDEO WE'LL USE TWO GRAPHING TECHNIQUES
TO SOLVE LINEAR EQUATIONS.
THE FIRST METHOD IS CALLED THE INTERSECTION METHOD.
IF A LINEAR EQUATION IS IN THE FORM OF F(X) IS = G(X),
FIRST WE'LL GRAPH Y1 = F(X) THEN WE'LL GRAPH Y2 = G(X).
SO ESSENTIONALLY WE'LL GRAPH THE RIGHT SIDE
AND THE LEFT SIDE OF THE EQUATION SEPARATELY.
AND THEN WE'LL FIND THE X-COORDINATE
OF THE POINT OF INTERSECTION.
THAT X-COORDINATE OF THE POINT OF INTERSECTION
WILL BE THE SOLUTION.
SO HERE'S THE IDEA.
CAN YOU TELL ON THIS GRAPH FOR WHICH X VALUE THEY ARE EQUAL?
SO HERE WE HAVE A GRAPH OF F(X). HERE WE HAVE A GRAPH OF G(X).
HERE'S THE POINT OF INTERSECTION
AND THE X-COORDINATE OF THIS POINT REPRESENTS THE SOLUTION.
SO X = 1 WOULD BE OUR SOLUTION.
SO WE WANTED TO SOLVE THESE TWO EQUATIONS
USING THE INTERSECTION METHOD,
THE LEFT WE'LL CALL F(X) AND THE RIGHT SIDE WE'LL CALL G(X).
WE'LL TYPE THIS INTO Y1.
WE'LL TYPE THIS INTO Y2
AND DETERMINE THE POINT OF INTERSECTION.
SO LET'S GET OUR GRAPHING CALCULATORS.
PRESS Y=.
ENTER IN THE LEFT SIDE,
REMEMBER YOUR SLOPE NEEDS TO BE IN A SET OF PARENTHESES.
THE RIGHT SIDE WILL BE Y2 AND WE'LL PRESS GRAPH.
HERE IS OUR POINT OF INTERSECTION.
THE X-COORDINATE OF THIS WILL BE OUR SOLUTION
TO THIS LINEAR EQUATION.
SO IF WE PRESS 2nd TRACE, WHICH BRINGS UP THE CALCULATION MENU,
OPTION 5,
THEN ENTER 3 TIMES,
THE X VALUE IS EQUAL TO 6 AT THAT POINT OF INTERSECTION,
THEREFORE OUR SOLUTION IS X = 6.
FOR THIS PROBLEM LET'S GO AHEAD AND PERFORM SUBSTITUTION
TO CHECK OUR ANSWER.
SO 2/3 x 6, THIS WOULD SIMPLIFY NICELY.
WE'D HAVE 4 - 5 = -12 + 11, WHICH DOES CHECK,
BECAUSE A -1 IS EQUAL TO -1.
SO THAT'S HOW YOU SOLVE A LINEAR EQUATION
USING THE INTERSECTION METHOD.
NOW IT MIGHT BE A BIG TIME SAVER IN A PROBLEM LIKE NUMBER 2
AS LONG AS YOU'RE CAUTIOUS
WHEN YOU ENTER IN F(X) AND G(X) INTO THE GRAPHING CALCULATOR.
SO AGAIN, THE LEFT SIDE WOULD BE F(X) OR WHAT WE CAN CALL Y1
AND THE RIGHT SIDE, OR WHAT WE CALL Y2 OR G(X).
LET'S GO BACK TO OUR GRAPHING CALCULATORS
AND SOLVE THIS EQUATION.
WE'LL CLEAR OUT OUR PREVIOUS FUNCTIONS.
NOW WHEN WE TYPE THIS IN, WE DO HAVE TO BE CAREFUL.
WE'RE GOING TO INCLUDE THE NUMBERATOR
IN A SET OF PARENTHESES.
SO WE HAVE THE LEFT SIDE IN Y1.
NOW WE'LL TYPE RIGHT SIDE INTO Y2.
THEN GRAPH.
SO THE X-COORDINATE OF THIS POINT OF INTERSECTION
REPRESENTS OUR SOLUTION.
IF WE PRESS 2nd TRACE, OPTION 5, ENTER 3 TIMES,
THIS X VALUE REPRESENTS OUR SOLUTION, X = 3.
SO AS YOU CAN SEE IN THIS PROBLEM,
IT IS A BIG TIME SAVER
BECAUSE IN ORDER TO SOLVE THIS LINEAR EQUATION,
IT WOULD TAKE MANY STEPS ALGEBRAICALLY,
BUT GRAPHICALLY, IT'S PRETTY STRAIGHTFORWARD.
ONE OF THE DRAWBACKS ON THIS METHOD
IS IF THE POINT OF INTERSECTION
DOES NOT OCCUR IN THAT STANDARD WINDOW,
YOU MAY SPEND QUITE A BIT OF TIME
TRYING TO LOCATE THAT POINT OF INTERSECTION.
BUT MOST OF THE PROBLEMS FROM OUR TEXT,
THE INTERSECTION POINT WILL OCCUR IN THAT STANDARD WINDOW.
LET'S GO AHEAD AND TAKE A LOOK AT OUR SECOND METHOD,
CALLED THE ZERO METHOD, FOR SOLVING EQUATIONS.
FOR THIS METHOD WHAT WE DO IS WE SET THE EQUATION EQUAL TO ZERO.
THE FORM WILL BE F(X) = 0 OR 0 = F(X).
THEN WE'LL GRAPH Y = F(X).
USING THIS METHOD WE LOOK FOR THE X-INTERCEPT,
ALSO CALLED THE ZERO.
THE X-INTERCEPT IS THE SOLUTION TO THE EQUATION.
SO WE DO HAVE TO PERFORM SOME ALGEBRA
TO SET THE EQUATION EQUAL TO ZERO.
BUT THEN AS SOON AS WE DO THAT, WE DO RESORT BACK TO GRAPHING.
SO HERE'S THE IDEA.
LOOK AT THE GRAPH OF Y = F(X).
CAN YOU FIND THE X-INTERCEPT OR WHERE Y = F(X) = 0?
WELL, THE LINE CROSSES THE X-AXIS AT X = 2.
THEREFORE, THE SOLUTION WOULD BE X = 2 IF THIS WAS OUR GRAPH.
SO ON THE INTERSECTION METHOD,
WE LOOK FOR A POINT OF INTERSECTION.
ON THE ZERO METHOD, WE LOOK FOR THE X-INTERCEPT.
LET'S TAKE A LOOK AT A COUPLE OF EXAMPLES OF THIS METHOD.
THE FIRST STEP IS WE DO HAVE TO SET IT EQUAL TO ZERO.
SO WE'LL ADD 36 ON BOTH SIDES,
SO THE RIGHT SIDE IS OUR F(X) OR OUR Y1.
SO WE'LL GO AHEAD AND GRAPH THIS ON OUR GRAPHING CALCULATOR
AND LOOK FOR THE X-INTERCEPT.
SO Y1 = 3X + 15.
HERE WE CAN PRETTY EASILY SEE THE X-INTERCEPT IS -5,
BUT THE CALCULATOR WILL FIND THAT FOR US
IF WE PRESS 2nd TRACE, AND THEN OPTION 2 FOR ZERO.
NOW WHEN WE CHOOSE THIS OPTION,
IT'S GOING TO ASK US FOR THE LEFT BOUND AND THE RIGHT BOUND.
THE LEFT BOUND TELLS US THAT WE MUST PLACE THE CURSOR
TO THE LEFT OF THE X-INTERCEPT.
AND IF WE'RE NOT SURE WHERE THE CURSOR IS,
IT TELL US THAT IT'S RIGHT HERE AT -2.3.
TO THE LEFT OF THIS X-INTERCEPT, IN THIS CASE, WOULD BE BELOW.
SO WE MOVE THAT BLINKING CURSOR SO THAT IT'S BELOW.
ANYWHERE BELOW IS FINE.
HIT ENTER.
NOW IT'S GOING TO ASK FOR THE RIGHT BOUND.
WE NEED TO BE TO THE RIGHT SIDE OF THIS POINT OF INTERSECTION.
WE NEED TO BE THE RIGHT SIDE OF THIS X-INTERCEPT.
SO IN THIS CASE, IT WOULD BE ABOVE.
HIT ENTER AND THEN ENTER ONE MORE TIME.
NOW WE CAN SEE THAT THE ZERO IS EQUAL TO -5,
WHICH REPRESENTS OUR SOLUTION.
WHEN WE'RE CHOOSING THE LEFT BOUND AND RIGHT BOUND,
WE'RE ESSENTIALLY TELLING THE CALCULATOR
WHERE TO LOOK FOR THE ZERO, OR THE X-INTERCEPT.
SO IN THIS CASE, X = -5.
LET'S GO AHEAD AND TRY IT ON NUMBER 4.
SO TO SET IT EQUAL TO ZERO, WE'LL SUBTRACT 3X ON BOTH SIDES.
AT THE SAME TIME WE'LL ALSO ADD 8 ON BOTH SIDES.
SO THE RESULT WOULD BE 2X + 2 = 0.
SO WE'LL GO BACK TO OUR GRAPHING CALCULATOR
AND GRAPH Y1 = 2X + 2.
OF COURSE NOW THAT WE HAVE IT IN THIS FORM,
YOU COULD PROBABLY SOLVE THIS EQUATION MENTALLY,
BUT LET'S GO AHEAD AND TAKE A LOOK AT THE GRAPH ANYWAY.
2X + 2, GRAPH. CAN'T SEE THE SOLUTION THERE.
LET'S GO AHEAD AND FIND IT USING THE TECHNOLOGY.
PRESS 2nd TRACE, OPTION 2, LEFT BOUND,
MEANS TO LEFT SIDE OF THE X-INTERCEPT,
SO SOMEWHERE OVER HERE WOULD BE OKAY.
HIT ENTER.
RIGHT SIDE OF THE X-INTERCEPT, ANYWHERE UP HERE WOULD BE OKAY.
PRESS ENTER AND THEN ENTER ONE MORE TIME.
WE HAVE A ZERO OF -1, WHICH REPRESENTS OUR SOLUTION.
SO THAT WRAPS UP HOW TO SOLVE LINEAR EQUATIONS GRAPHICALLY.
OF THE TWO METHODS, I PREFER THE INTERSECTION METHOD
BECAUSE YOU DON'T HAVE TO DO ANY ALGEBRAIC MANIPULATION
TO THE EQUATION
AND YOU CAN SEE USING THE ZERO METHOD,
YOU DO HAVE TO PERFORM SOME ALGEBRA FIRST,
WHICH MAY OR MAY NOT BE EASIER
THAN JUST SOLVING IT ALGEBRAICALLY TO BEGIN WITH.
I HOPE YOU FOUND THAT HELPFUL.