- WELCOME TO A LESSON ON DETERMINING
THE EQUATION OF A LINE IN SLOPE INTERCEPT FORM.
THE EQUATION Y = MX + B
IS CALLED THE SLOPE INTERCEPT FORM OF A LINE
WHERE M = THE SLOPE OF THE LINE
AND B = THE Y INTERCEPT OF THE LINE.
SO BASED UPON THE INFORMATION THAT WE'RE GIVEN,
IF WE CAN IDENTIFY THE SLOPE AND THE Y INTERCEPT
WE CAN THEN WRITE THE EQUATION OF THE LINE.
SO LOOKING AT OUR FIRST EXAMPLE
WE WANT TO DETERMINE THE EQUATION OF THE LINE
PASSING THROUGH THE POINT (0,2)
WITH THE SLOPE = -3/5 AND THEN WE WANT TO GRAPH THE LINE.
WELL NOTICE THE Y INTERCEPT OCCURS WHEN X = 0.
SO THIS POINT DOES REPRESENT THE Y INTERCEPT
AND WE'RE ALSO GIVEN THE SLOPE.
SO WE'RE GIVEN EVERYTHING WE NEED
TO WRITE THE EQUATION OF THE LINE.
B IS GOING TO BE = 2 AND M IS GOING TO BE = -3/5.
SO NOW WE'LL SUBSTITUTE -3/5 FOR M
AND 2 FOR B TO WRITE THE EQUATION OF THE LINE.
WE WOULD HAVE Y = -3/5X + 2.
AND NOW THEY ALSO ASK US TO GRAPH THIS LINE.
SO FROM HERE WE WOULD PLOT THE Y INTERCEPT, 2
AND THEN SINCE THE SLOPE IS -3/5,
FROM THE Y INTERCEPT WE'LL GO DOWN 3 UNITS
AND RIGHT 5 UNITS.
WE WENT DOWN 3 UNITS BECAUSE THE NUMERATOR WAS -3.
WE WENT RIGHT 5 UNITS BECAUSE THE DENOMINATOR WAS 5.
SO OUR LINE PASSES THROUGH THESE 2 POINTS
AND WOULD LOOK LIKE THIS.
LET'S TAKE A LOOK AT ANOTHER EXAMPLE.
HERE WE'RE GIVEN THE LINE THAT PASSES THROUGH THE POINT (4,-1)
AND HAS A SLOPE OF 2.
WELL NOTICE HOW THIS POINT IS NOT THE Y INTERCEPT
BECAUSE THE X COORDINATE IS NOT EQUAL TO ZERO.
SO WE KNOW THE LINE HAS A SLOPE OF 2
AND IT PASSES THROUGH THE POINT (4,-1).
SO FOR THIS PROBLEM WE CAN SUBSTITUTE 2 FOR M,
SO LET'S START WITH THAT.
WE KNOW THAT Y IS GOING TO BE = 2 x X + B.
NOW, WE STILL HAVE TO FIND THE Y INTERCEPT,
BUT SINCE WE ARE GIVEN ONE POINT ON THE LINE
WE CAN SUBSTITUTE 4 FOR X AND -1 FOR Y AND THEN SOLVE FOR B.
SO THIS WOULD GIVE US THE EQUATION -1 = 2 x 4 + B.
AND NOW THAT WE HAVE ONE UNKNOWN WE CAN SOLVE FOR B.
IF -1 = 8 + B SUBTRACT 8 ON BOTH SIDES THIS WOULD BE ZERO.
SO WE HAVE B = -9.
SO NOW WE'LL SUBSTITUTE -9 FOR B.
SO OUR EQUATION WOULD BE Y = 2X AND THEN INSTEAD OF WRITING + -9
WE CAN JUST WRITE - 9.
NOW THIS LINE IS GOING TO BE A LITTLE MORE DIFFICULT TO GRAPH
SINCE THE Y INTERCEPT = -9,
WHICH IS OFF THE SCREEN.
SO WHAT WE'LL DO IS PLOT THE GIVEN POINT (4,-1),
WHICH IS HERE AND THEN SINCE WE KNOW THE SLOPE IS = 2,
WHICH WE CAN WRITE AS 2/1.
FROM THIS POINT WE'LL MOVE UP TWO UNITS
AND WRITE ONE UNIT TO HERE.
AND OUR LINE WOULD PASS THROUGH THESE THREE POINTS
AND WOULD LOOK SOMETHING LIKE THIS.
NOW WE WANT TO FIND THE EQUATION OF THE LINE
PASSING THROUGH THE POINT (6,3) AND THE POINT (-3,-3).
SO NOTICE HOW NOW WE'RE NOT GIVEN THE SLOPE
AND NEITHER OF THESE POINTS ARE THE Y INTERCEPT.
BUT TO DETERMINE THE SLOPE OF A LINE PASSING THROUGH TWO POINTS
WE CAN USE THIS EQUATION HERE TO FIND THE SLOPE.
SO LET'S CALL THIS X SUB 1, Y SUB 1,
AND WE'LL CALL THIS X SUB 2 AND Y SUB 2.
SO THE SLOPE OF THE LINE
IS GOING TO BE EQUAL TO Y SUB 2 - Y SUB 1 IN THE NUMERATOR
AND X SUB 2 - X SUB 1 IN THE DENOMINATOR.
SO -3 - 3 IN THE NUMERATOR
AND THE DENOMINATOR IS GOING TO BE -3 - 6.
SO WE'RE GOING TO HAVE -6/-9
WHICH WILL SIMPLIFY TO 2/3.
SO NOW WE'LL SUBSTITUTE 2/3 FOR M
SO WE KNOW Y MUST = 2/3X + B.
AND NOW WE CAN FIND THE Y INTERCEPTOR B
BY USING ONE OF THE POINTS THAT WE KNOW IS ON THE LINE.
SO IF WE USE THE POINT (6,3) WHEN X IS 6, Y MUST BE 3.
SO WE'D HAVE THE EQUATION 3 = 2/3 x 6,
WHICH I'LL WRITE AS 6/1 + B.
THE 3 AND THE 6 SIMPLIFY THIS CHANGES TO 1--THIS CHANGES TO 2.
SO WE'D HAVE 3 = 2 x 2 = 4 + B.
SO NOW WE'LL SUBTRACT 4 ON BOTH SIDES TO SOLVE FOR B.
SO WE HAVE B = -1.
AND NOW THAT WE HAVE THE VALUE OF B
WE CAN WRITE THE EQUATION OF THE LINE,
IT'S GOING TO BE Y = 2/3X.
AND THEN AGAIN INSTEAD OF WRITING + -1
WE'LL JUST WRITE - 1.
NOW, TO GRAPH THIS LINE WE'LL PLOT THE Y INTERCEPT OF -1
AND THEN BECAUSE THE SLOPE IS 2/3 WE'LL GO UP 2 UNITS
AND RIGHT 3 UNITS TO HERE
AND OUR LINE PASSES THROUGH THESE TWO POINTS.
LET'S TAKE A LOOK AT ONE MORE EXAMPLE SAME TYPE OF PROBLEM.
WE'RE GIVEN TWO POINTS,
WE WANT TO FIND THE EQUATION OF THE LINE.
LET'S START BY DETERMINING THE SLOPE.
WE'LL CALL THIS X SUB 1, Y SUB 1, X SUB 2, Y SUB 2.
SO THE SLOPE IS GOING TO = Y SUB 2 - Y SUB 1.
WE'LL HAVE -3 - -3/X SUB 2 - X SUB 1.
THAT'LL BE 2 - -4.
WELL THIS ENDS UP BEING -3 + 3/2 + 4,
WHICH IS GOING TO BE 0/6 WHICH IS = 0.
NOW, WE SHOULD REMEMBER THAT IF THE SLOPE IS ZERO
THIS LINE IS GOING TO BE A HORIZONTAL LINE.
LET'S PLOT THESE 2 POINTS AND SEE WHAT THE LINE LOOKS LIKE,
(-4,-3) WOULD BE HERE AND THEN (2,-3) WOULD BE HERE.
SO WE HAVE A HORIZONTAL LINE PASSING THROUGH Y = -3.
IF WE GO BACK AND TAKE A LOOK AT THE 2 POINTS
THAT WERE GIVEN NOTICE HOW BOTH Y COORDINATES ARE = -3,
WHICH ACTUALLY TELLS US THE EQUATION OF OUR LINE IS Y = -3.
NOW, IF WE DIDN'T RECOGNIZE THIS
AND WE TRIED TO FIGURE OUT THE EQUATION IN THE NORMAL WAY
WE WOULD HAVE Y = 0X + B AND THEN WE'D USE
ONE OF THE POINTS TO SUBSTITUTE IN X AND Y.
SO IF WE USED THE FIRST POINT WE WOULD HAVE Y = -3
WHEN X = -4 AND NOTICE HOW THIS WOULD BE ZERO.
WE STILL HAVE B = -3 AND SINCE 0 x X = 0
WE WOULD HAVE THE EQUATION Y = -3.
SO EITHER WAY WE'D FIND THE EQUATION,
BUT AS SOON AS YOU RECOGNIZE THAT IT'S A HORIZONTAL LINE
WE KNOW WE'LL HAVE AN EQUATION IN THE FORM OF Y = A CONSTANT.
OKAY, I HOPE YOU FOUND THIS HELPFUL.