- WE WANT TO SOLVE THE SYSTEM OF LINEAR EQUATIONS
USING THE ELIMINATION METHOD OR THE ADDITION METHOD.
A SOLUTION TO A SYSTEM OF EQUATIONS IS AN ORDERED PAIR
OR AN X AND Y VALUE
THAT WOULD SATISFY BOTH OF THE GIVEN EQUATIONS.
AND THE IDEA BEHIND THE ELIMINATION METHOD
IS YOU WANT TO ADD THESE TWO EQUATIONS TOGETHER
AND WHEN DOING SO, ELIMINATE ONE OF THE VARIABLES.
THE ONLY WAY WE CAN ELIMINATE A VARIABLE
WHEN WE ADD THESE EQUATIONS TOGETHER
IS IF THE X TERMS OR Y TERMS ARE OPPOSITES.
SO ONCE THE EQUATIONS ARE IN STANDARD FORM,
AS WE SEE HERE WITH THE X AND Y TERMS ON THE LEFT
AND THE CONSTANTS ON THE RIGHT,
RIGHT AWAY WE SHOULD RECOGNIZED THAT THE X TERMS AND Y TERMS
ARE NOT OPPOSITES.
EITHER ONE OR BOTH EQUATIONS BY A CONSTANT SO THAT THEY ARE.
IF YOU LOOK AT THE X TERMS,
IF WE WANT THE X TERMS TO BE OPPOSITES,
SINCE WE HAVE 3X AND 7X WE'D HAVE TO HAVE A -21X AND +21X.
SO WE'D HAVE TO MULTIPLY THIS FIRST EQUATION BY EITHER 7 OR -7
AND THE SECOND EQUATION BY +3 OR -3.
SO WE'D HAVE TO MULTIPLY BOTH EQUATIONS BY A CONSTANT,
BUT IF YOU LOOK AT THE Y TERMS,
HERE WE HAVE +1Y AND HERE WE HAVE +5Y.
IF THIS WAS -5Y, THE Y TERMS WOULD BE OPPOSITES.
SO WE CAN MAKE THE Y TERMS OPPOSITES
IF WE MULTIPLY THE FIRST EQUATION BY -5.
LET'S GO AHEAD AND DO THAT.
WE'RE GOING TO MULTIPLY BOTH SIDES OF THE EQUATION BY -5,
WHICH WOULD GIVE US -15X - 5Y = +50.
BECAUSE THE Y TERM IS ALREADY + 5Y
WE'RE GOING TO LEAVE THE SECOND EQUATION THE SAME.
SO WE'LL HAVE 7X + 5Y = -18.
AND NOW WHEN WE ADD THESE TWO EQUATIONS TOGETHER
NOTICE HOW WE HAVE -5Y + 5Y.
THE Y TERMS WOULD HAVE A SUM OF ZERO.
THE SUM OF THE X TERMS WOULD BE -8X AND OVER HERE WE HAVE +32.
SO NOW WE CAN DIVIDE BOTH SIDES BY -8, AND NOW WE KNOW X = -4.
WE'RE NOT DONE, THOUGH,
BECAUSE REMEMBER THE SOLUTION IS AN ORDERED PAIR.
WE KNOW X = -4, AND NOW WE HAVE TO PERFORM SUBSTITUTION
INTO EQUATION ONE OR EQUATION TWO AND THEN SOLVE FOR Y.
LET'S GO AHEAD AND USE EQUATION ONE AND SUBSTITUTE -4 FOR X.
WE WOULD HAVE 3 x -4 + Y = -10.
SO THIS WOULD BE -12 + Y = -10.
SO WE'D ADD 12 TO BOTH SIDES.
AND HERE WE HAVE Y = +2.
SO OUR SOLUTION IS X = -4 AND Y = 2.
AND SINCE WE HAVE ONE UNIQUE SOLUTION,
WE CAN SAY THE SYSTEM IS CONSISTENT
BECAUSE IT HAS AT LEAST ONE SOLUTION
AND THE EQUATIONS ARE INDEPENDENT
BECAUSE THEY ARE DIFFERENT.
NOW LET'S GO AHEAD AND VERIFY OUR SOLUTION.
WE'LL SUBSTITUTE X = -4 AND Y = 2 INTO BOTH EQUATION ONE
AND EQUATION TWO.
SO FOR EQUATION ONE WE'D HAVE 3 x -4 + 2 = -10.
THIS WOULD BE -12 + 2 = -10.
SO THIS SATISFIES EQUATION ONE,
AND NOW LET'S CHECK EQUATION TWO.
WE WOULD HAVE 7 x -4 + 5 x 2 = -18.
SO WE HAVE -28 + 10 = -18, WHICH IS ALSO TRUE.
AND WE'LL GO AHEAD AND STOP HERE FOR THIS EXAMPLE.
I HOPE YOU FOUND IT HELPFUL.