- WE WANT TO DETERMINE THE EQUATION OF THE LINE
PASSING THROUGH THE GIVEN TWO POINTS.
WE WANT THE EQUATION TO BE IN SLOPE-INTERCEPT FORM.
SO TO DO THIS, WE CAN WORK WITH A SLOPE-INTERCEPT FORM
OR A POINT-SLOPE FORM OF A LINE.
IN THIS VIDEO, WE'LL ONLY BE USING SLOPE-INTERCEPT FORM
OF A LINE WHICH IS THE FORM Y = MX + B
WHERE M IS THE SLOPE OF THE LINE
AND B IS THE Y-INTERCEPT OF THE LINE.
AND SINCE WE'RE GIVEN TWO POINTS ON THE LINE,
WE CAN USE THE EQUATION M = Y SUB 2 - Y SUB 1
DIVIDED BY X SUB 2 - X SUB 1 TO FIND THE SLOPE.
SO IF WE CALL THESE COORDINATES
THE 1s AND THESE COORDINATES THE 2s,
THIS WOULD BE (X SUB 1, Y SUB 1),
AND THIS WOULD BE (X SUB 2, Y SUB 2).
SO NOW USING THIS SLOPE FORMULA, Y SUB 2 - Y SUB 1 = -2 - -3,
AND X SUB 2 - X SUB 1 = 4 - -2.
SO IF WE SIMPLIFY THIS, THIS BECOMES -2 + 3
AND THIS BECOMES 4 + 2, SO OUR SLOPE IS EQUAL TO 1/6.
SO LET'S GO AHEAD AND SUBSTITUTE 1/6 IN FOR M.
WE KNOW THAT OUR EQUATION MUST BE Y = 1/6X + B.
SO THE SECOND STEP IS GOING TO BE TO DETERMINE
THE VALUE OF B OR THE Y-INTERCEPT.
WELL IF WE LOOK BACK AT THE TWO POINTS THAT WERE GIVEN,
NEITHER OF THESE ARE THE Y-INTERCEPT,
BECAUSE NOTICE NEITHER OF THE X COORDINATES ARE EQUAL TO 0.
BUT THESE POINTS ARE ON THE LINE
WHICH MEANS IF WE SUBSTITUTE THE X AND Y COORDINATES
INTO THE LINEAR EQUATION,
IT MUST SATISFY THE LINEAR EQUATION.
SO WHAT WE CAN DO IS SELECT ONE OF THESE POINTS,
SUBSTITUTE THE X COORDINATE HERE
AND Y COORDINATE HERE WHICH WILL THEN ALLOW US TO SOLVE FOR B.
AND IT DOESN'T MATTER WHICH POINT WE USE.
LET'S GO AHEAD AND JUST USE THE SECOND POINT,
SO WE'LL SUBSTITUTE -2 FOR Y AND +4 FOR X.
SO WE'RE GOING TO HAVE -2 MUST EQUAL 1/6 x 4,
BUT I'LL WRITE 4 AS 4/1 + B.
AND NOW, WE CAN SOLVE THIS EQUATION FOR B.
WE'LL FIRST DETERMINE THIS PRODUCT HERE.
NOTE THERE'S A COMMON FACTOR OF 2.
THERE'S TWO 2s IN 4 AND THREE 2s IN 6, SO WE HAVE -2 EQUALS,
THIS WOULD BE 2/3 + B,
SO NOW WE'LL SUBTRACT 2/3 ON BOTH SIDES.
SO WE'D HAVE B EQUALS, THIS WOULD BE -2 2/3, OR IF WE WANT,
WE CAN CONVERT THIS TO AN IMPROPER FRACTION.
WE'D HAVE B EQUALS THE NEGATIVE FRACTION
WITH THE DENOMINATOR OF 3.
AND THEN FOR THE NUMERATOR,
WE'D HAVE 3 x 2 + 2 WHICH SHOULD BE 8.
SO NOW THAT WE KNOW THE VALUE OF B, USING THIS EQUATION HERE,
WE'LL SUBSTITUTE -8/3 FOR B.
SO THE EQUATION OF OUR LINE WOULD BE Y = 1/6X + -8/3,
BUT YOU NORMALLY WON'T SEE THE LINEAR EQUATION WRITTEN
IN THIS FORM, + -8/3 IS EQUIVALENT TO, SO TRACKING 8/3.
SO YOU'LL NORMALLY SEE THE EQUATION ON THE LINE WRITTEN
AS Y = 1/6X - 8/3.
LET'S GO AHEAD AND VERIFY THIS GRAPHICALLY.
NOTICE HOW THE Y-INTERCEPT IS - 8/3 OR -2 2/3 AS WE SEE HERE.
AND THEN TO VERIFY THE SLOPE,
IF WE SELECT TWO POINTS ON THE LINE WITH INTEGER COORDINATES,
FOR EXAMPLE, THIS POINT HERE AND THIS POINT HERE,
NOTICE HOW IF WE WANTED TO MOVE FROM THIS POINT TO THIS POINT,
WE'D HAVE TO GO UP ONE UNIT
AND RIGHT SIX UNITS WHICH DOES VERIFY THAT THE VERTICAL CHANGE
WOULD BE +1 AND THE HORIZONTAL CHANGE WOULD BE +6,
GIVING US A SLOPE OF +1/6.