– 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.

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