- WE WANT TO DETERMINE THE EQUATION OF A LINE
THAT IS PERPENDICULAR TO Y = 2X + 1,
AND PASSES THROUGH THE POINT WITH COORDINATES (-4,5).
IF TWO LINES ARE PERPENDICULAR,
THAT MEANS THEY INTERSECT AND FORM A RIGHT ANGLE
OR A 90 DEGREE ANGLE.
AND IF TWO LINES ARE PERPENDICULAR,
THEN THEIR SLOPES ARE NEGATIVE RECIPROCALS OF ONE ANOTHER,
WHICH CAN BE WRITTEN USING THIS NOTATION HERE,
WHICH CAN BE A LITTLE BIT CONFUSING.
SO FOR EXAMPLE, IF THIS FIRST LINE
HAD A SLOPE OF LET'S SAY, 2/5,
THEN THE SLOPE OF THE SECOND LINE HERE
WOULD HAVE TO BE THE NEGATIVE RECIPROCAL,
MEANING WE'LL FLIP THIS OVER AND THEN CHANGE THE SIGN.
SO IT'S GOING TO BE -5/2.
OKAY, SO BACK TO OUR PROBLEM,
WE WANT A LINE THAT IS PERPENDICULAR TO Y = 2X + 1.
WELL, THIS LINE IS IN SLOPE INTERCEPT FORM,
SO WE SHOULD RECOGNIZE THAT THE SLOPE OF THIS LINE IS 2/1.
THEREFORE, THE SLOPE OF THE PERPENDICULAR LINE
WOULD BE THE NEGATIVE RECIPROCAL OF THIS.
SO FOR THE SLOPE OF OUR LINE,
WE'LL HAVE TO FLIP THIS OVER AND CHANGE THE SIGN.
THAT WOULD GIVE US -1/2.
-1/2 IS A NEGATIVE RECIPROCAL OF 2/1.
AND THEN WE KNOW OUR LINE ALSO PASSES THROUGH THE POINT (-4,5).
NOW, TO DETERMINE THE EQUATION OF THE LINE WITH SLOPE OF -1/2
CONTAINING THE POINT (-4,5),
WE CAN EITHER USE SLOPE INTERCEPT FORM,
OR THE POINT SLOPE FORM OF THE LINE.
AND I THINK FOR THIS VIDEO WE'LL SHOW BOTH.
SO USING JUST SLOPE INTERCEPT FORM OR THE FORM Y = MX + B,
WE WOULD FIRST SUBSTITUTE -1/2 FOR M.
SO WE'D HAVE Y = -1/2X + B.
AND THEN TO DETERMINE THE Y INTERCEPT,
IT'S NOT GOING TO BE +5 BECAUSE THE X COORDINATE IS NOT ZERO.
BUT SINCE THIS POINT IS ON THE LINE
THAT MEANS IF WE SUBSTITUTE THESE COORDINATES
INTO THE EQUATION
IT SHOULD SATISFY THE EQUATION.
SO WE'LL SUBSTITUTE -4 FOR X, 5 FOR Y, AND THEN SOLVE FOR B.
SO WE'D HAVE 5 = -1/2 x -4 + B.
PUT THAT -4/1 SO WE HAVE 5 EQUALS--
AND SO THIS SIMPLIFIES HERE, THERE'S ONE 2 IN 2,
AND TWO 2s IN 4.
THIS BECOMES -1 x -2.
DON'T FORGET ABOUT THE SIGN HERE.
SO NOW IF WE SUBTRACT 2 ON BOTH SIDES--
WE HAVE B EQUALS-- THIS WILL BE +3.
SO NOW WE HAVE OUR EQUATION, AND WE'LL JUST SUBSTITUTE +3 FOR B.
AND THE EQUATION WILL BE Y = -1/2X + 3.
AND LET'S GO AHEAD AND SEE IF WE CAN GET THE SAME EQUATION
USING POINT SLOPE FORM.
POINT SLOPE FORM OF A LINE IS Y - Y1 = M(X - X SUB 1),
WHERE M IS THE SLOPE,
AND (X SUB 1,Y SUB 1) IS A POINT ON THE LINE.
SO FOR THIS SITUATION X SUB 1 IS GOING TO BE -4,
AND Y SUB 1 WILL BE +5.
SO WE'D HAVE Y - 5 = -1/2(X).
NOW, HERE WE'D HAVE MINUS -4, WHICH IS THE SAME AS +4.
AND NOW WE'LL GO AHEAD AND SOLVE THIS EQUATION FOR Y.
SO WE'LL DISTRIBUTE HERE,
SO WE'LL HAVE Y - 5 = THIS WOULD BE -1/2X.
THEN HERE WE'D HAVE -2,
AND THE LAST STEP TO SOLVE FOR Y WOULD BE TO ADD 5.
THIS WOULD BE 0, SO WE'D HAVE Y = -1/2X,
AND THIS WILL BE +3.
SO NOTICE HOW IF WE JUST USE SLOPE INTERCEPT FORM
TO DETERMINE THE EQUATION,
OR IF WE USE POINT SLOPE FORM TO DETERMINE THE EQUATION,
THE RESULT IS THE SAME.
AND LET'S GO AHEAD AND QUICKLY VERIFY THIS GRAPHICALLY.
YOU WERE GIVEN THE RED LINE Y = 2X + 1,
AND WE DETERMINED THAT THE EQUATION
OF THE PERPENDICULAR LINE WAS Y = -1/2X + 3.
NOTICE HOW THESE TWO LINES DO INTERSECT IN A RIGHT ANGLE HERE,
AND THE BLUE LINE DOES CONTAIN THE GIVEN POINT
THAT HAD A COORDINATES (-4,5), THIS POINT HERE.
SO OUR WORK LOOKS GOOD.
I HOPE YOU FOUND THIS HELPFUL.