- WELCOME TO AN EXAMPLE ON HOW TO SOLVE A SYSTEM
OF NONLINEAR EQUATIONS.
WE WANT TO FIND ALL THE SOLUTIONS TO THE SYSTEM
Y = X SQUARED - 9 AND Y = 9 - X SQUARED.
SO WE'LL SOLVE THIS SYSTEM ALGEBRAICALLY
USING ELIMINATION OR SUBSTITUTION
AND THEN VERIFY OUR SOLUTION GRAPHICALLY.
BECAUSE BOTH OF OUR EQUATIONS ARE SOLVED FOR Y,
WE'LL GO AHEAD AND USE SUBSTITUTION.
SO LOOKING AT THE FIRST EQUATION, IF Y = X SQUARED - 9,
WE CAN SUBSTITUTE X SQUARED - 9 FOR Y IN THE SECOND EQUATION.
SO THIS WOULD GIVE US X SQUARED - 9 = 9 - X SQUARED.
AND NOW WE WANT TO SOLVE THIS EQUATION FOR X
AND WE'LL PERFORM BACK SUBSTITUTION
TO FIND THE VALUE OF Y.
SO WE FIRST WANT TO HAVE THE X SQUARED TERMS
ON ONE SIDE OF THE EQUATION,
SO WE'LL ADD X SQUARED TO BOTH SIDES.
THIS WILL GIVE US 1X SQUARED + 1X SQUARED
OR 2X SQUARED - 9 = 9.
THESE TWO TERMS ARE OPPOSITES, SO THEY SIMPLIFY OUT.
SO FOR THE NEXT STEP, WE'LL ADD 9 TO BOTH SIDES OF THE EQUATION.
THIS WOULD GIVE US 2X SQUARED,
THESE TWO TERMS SIMPLIFY OUT, = 18.
THE NEXT STEP, WE'LL DIVIDE BOTH SIDES BY 2.
SO NOW WE HAVE X SQUARED = 9.
NOW WE SHOULD RECOGNIZE FOR THIS EQUATION,
WE'LL HAVE TWO SOLUTIONS BECAUSE X IS SQUARED.
SO TO UNDO THE SQUARING ON X,
WE'LL TAKE THE SQUARE ROOT OF BOTH SIDES OF THE EQUATION,
AND THEN WE'LL ALSO INCLUDE A + OR - SIGN HERE TO MAKE SURE
WE OBTAIN BOTH SOLUTIONS.
SO WE HAVE X = + OR - 3 OR X = + 3 OR -3.
THIS IS NORMALLY HOW MOST TEXTS SHOW HOW TO SOLVE THIS EQUATION.
BUT THE REAL REASON WHY WE HAVE TWO SOLUTIONS HERE
IS BECAUSE WHEN WE TAKE THE SQUARE ROOT OF X SQUARED,
IT'S ACTUALLY EQUAL TO THE ABSOLUTE VALUE OF X,
BECAUSE THE RESULT MUST BE POSITIVE.
SO WE HAVE THE ABSOLUTE VALUE OF X = +3,
THEREFORE X CAN BE EITHER BOTH +3 OR -3.
SO NOW THAT WE KNOW X CAN BE EITHER + OR -3,
WE NEED TO PERFORM BACK SUBSTITUTION
INTO ONE OF THE ORIGINAL EQUATIONS
TO FIND THE CORRESPONDING Y VALUES.
LET'S GO AHEAD AND USE THE FIRST EQUATION WHERE WE HAVE
Y = X SQUARED - 9.
LET'S START WITH X = +3, SO WE WOULD HAVE Y = +3 SQUARED -9,
SO Y = 9 - 9 = 0 WHICH MEANS WHEN X = +3, Y IS 0.
BECAUSE X IS SQUARED WHEN X IS -3,
WE'LL HAVE THE SAME CORRESPONDING Y VALUE.
LET'S GO AHEAD AND SHOW IT.
IF X = -3, THEN WE HAVE Y = -3 SQUARED - 9,
BUT -3 SQUARED IS STILL 9, SO 9 - 9 = 0.
FOR OTHER SOLUTION IS X = -3 AND Y = 0.
SO THE SYSTEM OF NONLINEAR EQUATIONS HAS TWO SOLUTIONS
WHICH MEANS, IF YOU GRAPH THESE TWO EQUATIONS
ON THE COORDINATE PLANE,
THE TWO FUNCTIONS WHICH ARE PARABOLAS SHOULD INTERSECT
AT THESE TWO POINTS.
LET'S GO AHEAD AND VERIFY THAT.
NOTICE ONE PARABOLA OPENS UP AND ONE OPENS DOWN.
THERE ARE TWO POINTS OF INTERSECTIONS,
BOTH ON THE X-AXIS HERE AND HERE,
AND NOTICE THE COORDINATES OF THE POINT OF INTERSECTION
ARE (3,0) AND (-3,0) WHICH WERE THE SOLUTIONS
THAT WE FOUND ALGEBRAICALLY.
OKAY. I HOPE YOU FOUND THIS EXPLANATION HELPFUL.
THANK YOU FOR WATCHING.