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- WELCOME TO A VIDEO ON CRAMER'S RULE.
THIS IS A WAY OF SOLVING A SYSTEM OF EQUATIONS
USING DETERMINANTS.
LET'S FIRST TAKE A LOOK AT A SYSTEM OF TWO EQUATIONS
WITH TWO UNKNOWNS.
NOTICE THE (A) ARE THE COEFFICIENT OF THE X TERMS.
THE (B) ARE THE COEFFICIENT OF THE Y TERMS
AND THE (C) ARE THE CONSTANTS.
SO IF THE SYSTEM IS SET UP THIS WAY,
X IS EQUAL TO THIS QUOTIENT AND Y IS EQUAL TO THIS QUOTIENT.
IF YOU FIRST TAKE A LOOK AT THE DENOMINATOR OF THESE QUOTIENTS
THE ELEMENTS IN THE DETERMINANT ARE FORMED BY THE COEFFICIENTS
OF THE X AND Y TERMS.
IF WE TAKE A LOOK AT THE NUMERATOR FOR THE VALUE OF X.
IF WE REPLACE THE X COEFFICIENTS WITH THE CONSTANTS
WE HAVE THE NUMERATOR FOR THE VALUE OF X.
SIMILARLY FOR Y, IF WE TAKE OUT THE Y COEFFICIENTS
AND REPLACEMENT WITH THE CONSTANTS
WE HAVE THE NUMERATOR FOR THE VALUE OF Y.
LET'S GO AHEAD AND TAKE A LOOK AT AN EXAMPLE.
LET'S GO AHEAD AND SEE IF WE CAN SOLVE THIS USING DETERMINANTS.
WE KNOW THE VALUES OF X AND Y WILL BE A QUOTIENT
OF A DETERMINANT.
SO LET'S GO AHEAD AND SET UP OUR QUOTIENTS.
NEXT, THE DETERMINANT AND THE DENOMINATOR
WILL BE THE 2 BY 2 DETERMINANT FORMED BY THE COEFFICIENT
OF THE X AND Y TERMS.
SO THE FIRST ROW WILL BE (1, -1)
AND THE SECOND ROW WILL BE (1, 4) BOTH FOR X AND Y.
NOW THE NUMERATOR FOR THE VALUE OF X WILL COME FROM
REPLACING THE X COEFFICIENTS WITH THE CONSTANT TERMS.
SO WE'LL TAKE OUT THE 1 AND THE 1
AND REPLACE THEM WITH -3 AND 17
AND THIS COLUMN STAYS THE SAME.
FOR Y WE'RE GOING TO REPLACE THE Y COEFFICIENTS
WITH THE CONSTANTS.
SO WE'LL TAKE OUT THE -1 AND 4 AND REPLACE IT WITH -3 AND 17
AND THE X COLUMN STAYS THE SAME.
LET'S GO AHEAD AND EVALUATE THESE DETERMINANTS.
HERE WE'LL HAVE -12 MINUS -17 THAT WILL BECOME -12 + 17,
THAT'S 5.
THE DENOMINATOR WILL BE 4 MINUS -1 OR 4 +1, THAT'S 5.
AND REMEMBER THE DENOMINATORS ARE THE SAME
SO THIS WILL ALSO BE 5.
AND THE NUMERATOR FOR THE Y VALUE WILL BE 17 MINUS -3
OR 17 +3, THAT WILL BE 20.
SO THE SOLUTION TO THE SYSTEM WILL BE X = 1 AND Y = 4.
LET'S GO AHEAD AND TAKE A LOOK AT A SYSTEMS OF THREE EQUATIONS
AND THREE VARIABLES.
NOW, IT LOOKS LIKE A LOT IS GOING ON HERE
BUT THE PATTERN DOES REMAIN THE SAME FOR THIS TYPE OF SYSTEM.
AND WHAT I MEAN BY THAT IS IF YOU LOOK AT THE DENOMINATORS
OF EACH OF THESE VARIABLES
IT'S A DETERMINANT FORMED BY THE COEFFICIENTS
OF THE X, Y AND Z TERMS.
NEXT, IF YOU WANT TO SOLVE FOR THE X VALUE
THE NUMERATOR WILL COME FROM REPLACING THE X COEFFICIENTS
WITH THE CONSTANTS.
THE NUMERATOR FOR Y WILL COME FROM REPLACING
THE Y COEFFICIENTS WITH THE CONSTANTS.
AND THE NUMERATOR FOR Z WILL COME FROM REPLACING
THE Z COEFFICIENTS WITH THE CONSTANTS.
SO LET'S GO AHEAD AND TRY ONE OF THESE AS WELL.
SO WE'LL HAVE X, Y AND Z.
EACH WILL BE A QUOTIENT OF DETERMINANTS.
SO EACH OF THE DETERMINANTS HAVE BEEN DONE.
THEY WILL COME FROM THE COEFFICIENTS
OF THESE THREE EQUATIONS.
SO THE FIRST ROW WILL BE (2, 3, 1), (-1, 2, 3)
AND (-3, -3, 1).
REMEMBER, EACH OF THESE DENOMINATORS WILL BE THE SAME.
NOW, THE NUMERATOR FOR X WILL COME FROM THE DETERMINANT
FORMED BY REPLACING THE X COEFFICIENTS
WITH THE CONSTANTS TO (-1, 0)
SO WE'LL HAVE (2, -1, 0)
AND THE NEXT TWO COLUMNS WILL STAY THE SAME.
THE NUMERATOR FOR Y WILL COME FROM REPLACING
THE Y COEFFICIENTS WITH THE CONSTANTS TO (-1, 0).
COLUMN ONE AND COLUMN THREE WILL STAY THE SAME.
AND FOR THE NUMERATOR OF Z WE WILL REPLACE THE Z COEFFICIENTS
WITH (2, -1 AND 0)
AND THE FIRST TWO COLUMNS STAY THE SAME.
- LET'S GO AHEAD AND EVALUATE THESE
ON THE GRAPHING CALCULATOR.
SO WE'LL PRESS SECOND, MATRIX, GO OVER TO EDIT, PRESS ENTER.
LET'S GO AHEAD AND CALCULATE THE DENOMINATORS
OF ALL OF THESE FIRST.
- LET'S GO BACK TO THE HOME SCREEN.
LET'S FIND THE VALUE OF THESE DENOMINATORS.
SO WE'LL FIND THE DETERMINANT OF MATRIX A,
SECOND MATRIX, MATH, ENTER, SECOND MATRIX A,
CLOSE THAT AND IT'S EQUAL TO 7.
SO EACH OF THESE DENOMINATORS IS EQUAL TO 7.
- NOW LET'S GO AHEAD AND EVALUATE EACH OF THE NUMERATORS.
LET'S GO AHEAD AND PUT THE NUMERATORS IN
AS MATRIX B, C AND D.
- SO (2, 3, 1),
(-1, 2, 3),
(0, -3, 1).
- LET'S GO AHEAD AND ENTER THIS IN AS MATRIX C.
- 3 BY 3 SO WE HAVE (2, 2, 1), (-1, -1, 3).
- AND (-3, 0, 1).
- AND NOW LET'S GO AHEAD AND ENTER IN MATRIX D.
- LET'S GO AHEAD AND MOVE THE CALCULATOR FOR THAT ONE.
THE FIRST ROW IS (2, 3, 2),
(-1, 2, -1),
(-3, -3, 0).
NOW, LET'S GO AHEAD AND DETERMINE THE NUMERATORS
OF THESE VALUES.
SO REMEMBER THE NUMERATOR FOR X
WOULD BE THE DETERMINANT OF MATRIX B.
SO THE SECOND MATRIX, MATH, ENTER,
MATRIX B SO SECOND MATRIX B.
WE HAVE 28.
LET'S GO AHEAD AND FIND THE DETERMINANT OF MATRIX C.
SECOND MATRIX, MATH, ENTER, SECOND MATRIX.
THAT WOULD BE MATRIX C.
- THAT'S -21.
AND THEN FOR THE NUMERATOR OF Z WE'LL HAVE THE DETERMINANT OF...
- WHICH IS EQUAL TO 21.
SO WE'LL HAVE (28, -21 AND 21).
- THESE ALL COME OUT VERY NICE.
X IS EQUAL TO 4,
Y IS EQUAL TO -3
AND Z IS EQUAL TO +3.
SO CRAMER'S RULE WOULD BE QUITE A BIT OF WORK TO DO BY HAND
BUT IF YOU'RE ALLOWED TO USE TECHNOLOGY
IT'S A GREAT WAY TO USE WHAT WE'VE LEARNED ABOUT MATRIXES IS
AND DETERMINANTS TO SOLVE A SYSTEM OF EQUATIONS.
THANK YOU FOR WATCHING.