- A LINE HAS A SLOPE OF -2/3 AND PASSES THROUGH THE POINT
WITH COORDINATES -4, 6.
WE'RE GOING TO WRITE THE LINE IN POINT SLOPE FORM
AS WELL AS SLOPE INTERCEPT FORM.
IN POINT SLOPE FORM,
M IS THE SLOPE OF THE LINE AND X SUB 1 Y SUB 1
IS ANY POINT ON THAT LINE AND THEN FOR SLOPE INTERCEPT FORM,
M IS THE SLOPE OF THE LINE AND B IS THE Y INTERCEPT OF THE LINE.
SO LOOKING AT THE GIVEN INFORMATION,
WE KNOW M IS = -2/3
AND THEN WE'RE GIVEN THE POINT WITH COORDINATES -4, 6.
SO WE CAN LET THE FIRST COORDINATE
BE X SUB 1 FOR POINT SLOPE FORM AND WE CAN LET 6 = Y SUB 1.
SO THE EQUATION OF THE LINE IS POINT SLOPE FORM
WOULD BE Y - 6 = -2/3 x X - -4.
AND WE DON'T WANT TO LEAVE THE EQUATION IN THIS FORM.
REMEMBER SUBTRACTING A NEGATIVE IS THE SAME
AS ADDING A POSITIVE.
SO LET'S GO AHEAD AND REWRITE THIS AS
Y - 6 = -2/3 x THE QUANTITY X + 4.
NOW TO DETERMINE THE LINE IN SLOPE INTERCEPT FORM
WE CAN USE THE POINT SLOPE FORM OF THE EQUATION
AND THEN SOLVE FOR Y.
SO LET'S GO AHEAD AND REWRITE THIS.
WE'D HAVE Y - 6 = -2/3 x THE QUANTITY X + 4.
SO LET'S START BY DISTRIBUTING HERE.
YOU WOULD HAVE Y - 6 = THIS WOULD BE -2/3X
AND THEN -2/3 x 4 OR WE CAN THINK OF THIS AS 4/1.
WE'RE GOING TO HAVE - 8/3.
AND TO SOLVE FOR Y WE'LL ADD 6 TO BOTH SIDES.
SO WE HAVE 6 HERE AND LET'S PUT 6 HERE.
SO ON THE LEFT SIDE WE HAVE Y = THIS WOULD BE -2/3X
AND THEN HERE WE HAVE -8/3 + 6.
LET'S TAKE A CLOSER LOOK AT THIS OVER HERE ON THE SIDE.
-8/3 + 6 OR 6/1 WE KNOW WE HAVE TO HAVE A COMMON DENOMINATOR
WHICH WOULD BE 3 SO WE'LL MULTIPLY 6/1 BY 3/3
AND SO NOW WE'D HAVE -8/3 PLUS THIS WOULD BE 18/3
WHICH WOULD BE 10/3, WHICH IS OUR Y INTERCEPT.
SO WE HAVE 10/3.
SO HERE'S THE EQUATION IN SLOPE INTERCEPT FORM.
NOW LET'S VERIFY THESE TWO EQUATIONS
BY TAKING A LOOK AT THE GRAPH ON THE LINE.
LET'S START BY VERIFYING THE SLOPE
SO WE'LL SELECT TWO POINTS ON THIS LINE
WITH INTEGER COORDINATES.
FOR EXAMPLE, WE COULD SELECT
THIS POINT HERE AND THIS POINT HERE.
SO IF WE WANTED TO MOVE FROM THIS POINT TO THIS POINT
NOTICE HOW WE'D HAVE TO GO DOWN 2 UNITS AND RIGHT 3 UNITS
SO THE VERTICAL CHANGE IS -2 AND THE HORIZONTAL CHANGE IS 3.
WELL THIS VERIFIES OUR SLOPE IS CORRECT.
-2/3 HERE AND -2/3 HERE.
AND THEN TO VERIFY THE SLOPE INTERCEPT FORM,
NOTICE HOW THE Y INTERCEPT IS 10/3 WHICH IS 3 1/3
WHICH IS OUR Y INTERCEPT RIGHT HERE,
AND THEN TO VERIFY THE POINT SLOPE FORM OF THE EQUATION
WE'D RECOGNIZE THAT THIS IS TELLING US
THAT THIS CONTAINS A POINT WITH AN X COORDINATE OF -4
AND A Y COORDINATE OF 6.
SO -4, 6 IS RIGHT HERE ON OUR LINE.
SO THAT EQUATION IS ALSO CORRECT.
OKAY. I HOPE YOU FOUND THIS HELPFUL.