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Ex: Solve a System of Equations Using Substitution – Infinite Solutions


 – WE WANT TO SOLVE THE SYSTEM
OF EQUATIONS USING THE SUBSTITUTION METHOD, WHICH IS AN ALGEBRAIC METHOD FOR
SOLVING A SYSTEM OF EQUATIONS. TO SOLVE A SYSTEM OF EQUATIONS
WE WANT TO FIND AN ORDERED PAIR OR AN X AND Y VALUE THAT WOULD
SATISFY BOTH OF THE EQUATIONS, BUT WE DO HAVE TO KEEP IN MIND THAT IT IS POSSIBLE THAT THERE
MAY BE NO SOLUTION OR EVEN AN INFINITE NUMBER
OF SOLUTIONS TO A SYSTEM OF EQUATIONS. TO KEEP THINGS ORGANIZED
LET’S CALL THIS EQUATION ONE AND WE’LL CALL
THIS EQUATION TWO. THE FIRST STEP
IN THE SUBSTITUTION METHOD IS TO SOLVE ONE OF THE EQUATIONS
FOR ONE OF THE VARIABLES. LUCKILY, IF WE TAKE A LOOK AT
EQUATION TWO IT’S ALREADY SOLVED FOR Y. WE HAVE Y=5/2X – 5. THE NEXT STEP
IN THE SUBSTITUTION METHOD IS TO USE THIS EQUATION HERE
THAT’S SOLVED Y AND PERFORM SUBSTITUTION
INTO EQUATION ONE, MEANING IF WE KNOW THAT
Y=5/2X – 5, WE CAN REPLACE Y IN THE FIRST
EQUATION WITH 5/2X – 5. DOING THIS WILL GIVE US AN
EQUATION WITH ONE VARIABLE. SO BY PERFORMING THIS
SUBSTITUTION, EQUATION 1 WOULD BE 5X – 2 x,
INSTEAD OF Y, WE’LL HAVE 2 x 5/2X – 5=10. AGAIN, NOTICE HOW WE HAVE AN
EQUATION WITH ONE VARIABLE, IN THIS CASE X. SO NOW WE’LL SOLVE THIS EQUATION
FOR X AND THEN PERFORM BACK
SUBSTITUTION TO DETERMINE THE VALUE OF Y. SO THE FIRST STEP HERE IS GOING
TO BE TO CLEAR THE PARENTHESIS. SO WE’LL DISTRIBUTE. SO WE’LL HAVE 5X. FOR THIS FIRST PRODUCT WE COULD
THINK OF THIS AS -2/1. NOTICE HOW THE 2s WOULD SIMPLIFY
OUT SO WE’D HAVE -5X. AND THEN FOR SECOND PRODUCT
WE CAN THINK OF THIS AS -2 x -5=+10. SO WE’D HAVE +10=10. NOTICE HOW WE DO HAVE LIKE TERMS
HERE AND 5X – 5X=0, SO WE’RE LEFT WITH 10=10. AND NOTICE HOW THERE IS NO
LONGER ANY VARIABLE TERMS. WE’RE LEFT WITH 10=10. WHEN THE VARIABLES SIMPLIFY OUT AND WE’RE LEFT WITH A TRUE
STATEMENT OR TRUE EQUATION, 10 IS ALWAYS EQUAL TO 10, THIS TELLS US WE HAVE AN
INFINITE NUMBER OF SOLUTIONS. IF THE VARIABLES SIMPLIFY OUT AND WE’RE LEFT WITH A FALSE
STATEMENT OR FALSE EQUATION, WE WOULD HAVE NO SOLUTIONS. IF YOU REMEMBER WHEN WE WERE
SOLVING EQUATIONS GRAPHICALLY, IF A SYSTEM HAD AN INFINITE
NUMBER OF SOLUTIONS THE TWO EQUATIONS PRODUCED
THE SAME LINE. SO LET’S GO AHEAD
AND TAKE A MOMENT AND LOOK AT THE GRAPH OF THESE
TWO EQUATIONS. WELL EQUATION ONE AND EQUATION
TWO DO PRODUCE THE SAME LINE, AND THEREFORE THEY HAVE
AN INFINITE NUMBER OF POINTS IN COMMON, WHICH DOES VERIFY THAT WE HAVE AN INFINITE NUMBER
OF SOLUTIONS.  

Bernard Jenkins

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