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- WE'RE GOING TO DETERMINE THE EQUATION OF THE LINE
PASSING THROUGH THE POINTS (3,1) AND (5,-1).
WE WANT THE EQUATION TO BE IN SLOPE INTERCEPT FORM.
WHERE SLOPE INTERCEPT FORM IS Y = MX + B,
AND M IS THE SLOPE OF THE LINE,
AND B IS THE Y INTERCEPT OF THE LINE.
NOW, WE COULD USE POINT SLOPE FORM OF A LINE
TO HELP US DETERMINE THIS EQUATION,
WHICH WE WILL DO IN A FUTURE VIDEO.
BUT RIGHT NOW WE'RE ONLY GOING TO BE USING
SLOPE INTERCEPT FORM.
SO WE'LL START BY DETERMINING THE SLOPE.
REMEMBER IF WE HAVE TWO POINTS ON A LINE,
WE CAN DETERMINE THE SLOPE OF THE LINE
BY USING THE FORMULA M = Y SUB 2 - Y SUB 1
DIVIDED BY X SUB 2 - X SUB 1,
OR THESE ARE THE Y COORDINATES AND THESE ARE THE X COORDINATES.
SO LET'S START BY FINDING THE SLOPE.
LET THESE BE THE ONEs, AND THESE BE THE TWOs,
SO THIS WILL BE X SUB 1, Y SUB 1, X SUB 2, AND Y SUB 2.
SO WE HAVE M = (-1 - 1)/(5 - 3).
SO WE'RE GOING TO HAVE -2 DIVIDED BY +2,
WHICH IS EQUAL TO -1.
SO NOW THAT WE KNOW THE SLOPE,
WE KNOW THAT Y MUST EQUAL -1X + B.
AND WE'LL SIMPLIFY -1X LATER.
NOW LET'S DETERMINE THE Y INTERCEPT.
LOOKING AT THE TWO GIVEN POINTS,
NETHER OF THESE ARE THE Y INTERCEPTS,
BECAUSE NOTICE THE X COORDINATES ARE NOT ZERO.
BUT SINCE THESE TWO POINTS ARE ON THE LINE,
THAT MEANS THESE COORDINATES MUST SATISFY THE EQUATION.
MEANING IF WE SUBSTITUTE ONE OF THE X VALUES HERE
AND THE CORRESPONDING Y VALUE HERE,
THE LEFT SIDE MUST EQUAL THE RIGHT SIDE,
WHICH WILL ALLOW US TO DETERMINE B.
SO LET'S DETERMINE THE VALUE OF B BY USING THIS FIRST POINT HERE
WITH COORDINATES (3,1).
SO WE'LL SUBSTITUTE 3 FOR X AND 1 FOR Y.
SO WE WOULD HAVE +1 MUST EQUAL -1 x 3 + B.
AND SINCE WE HAVE ONE UNKNOWN, WE CAN NOW SOLVE FOR B.
WE HAVE 1 = -3 + B,
SO WE'LL ADD 3 TO BOTH SIDES OF THE EQUATION.
SO WE HAVE B = 4, WHICH IS OUR Y INTERCEPT.
SO NOW WE CAN SUBSTITUTE 4 FOR B,
AND THIS WOULD BE THE EQUATION OF THE LINE.
WE WOULD HAVE Y = -1 x X + 4.
AND NORMALLY YOU WON'T SEE THE EQUATION LEFT IN THIS FORM.
-1 x X WOULD JUST BE -X.
SO LET'S GO AHEAD AND WRITE THIS AGAIN AS Y = -X + 4.
NOW LET'S TAKE A MOMENT AND VERIFY THIS GRAPHICALLY.
YOU WERE GIVEN THESE TWO POINTS.
THIS POINT HAD THE COORDINATES (3,1),
AND THIS POINT HAD THE COORDINATES (5,-1).
AND WE FOUND THE EQUATION OF THIS LINE TO BE Y = -X + 4,
WHICH REMEMBER IS THE SAME THING AS Y = -1X + 4.
SO BY LOOKING AT THIS LINE LET'S QUICKLY VERIFY
THAT THE SLOPE IS EQUAL TO -1 OR -1/1,
AND THE Y INTERCEPT IS +4.
WELL, THIS BLUE LINE PASSING THROUGH THE GIVEN TWO POINTS
DOES HAVE A Y INTERCEPT OF +4.
AND THEN NOTICE IF WE SELECT ANY POINT ON THIS LINE
WITHIN INTEGER COORDINATES,
LET'S SAY THIS POINT HERE,
IF WE MOVE DOWN ONE UNIT AND RIGHT ONE UNIT,
WE DO END UP BACK ON THE BLUE LINE.
AND NOTICE THAT WE HAD A VERTICAL CHANGE OF -1
AND A HORIZONTAL CHANGE OF +1,
WHICH DOES VERIFY THE SLOPE WOULD -1/1 OR JUST -1.
SO THIS DOES VERIFY THAT OUR EQUATION IS CORRECT.