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3 years ago

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How many solutions does this system of equations have? Use either the substitution method or the elimination method. −2x + 4y =

4−2x + 4y = 4 x − 2y = 6x − 2y = 6 exactly one solution no solution infinitely many solutions

1 answer:

Katen [24]3 years ago

7 0

-2x+4y=4 Equation 1
x-2y=6 Equation 2
Solving by substitution method.
Isolate x from equation 2.
x=2y+6
Substitute value of x in equation 1
-2(2y+6)+4y=4
-4y-12+4y=4
-12=4
-16=0
Which is false.

Answer: No solution

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i. The Lagrangian is

$L(x,y,\lambda)=81x^2+y^2+\lambda(4x^2+y^2-9)$

with critical points whenever

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We have $f(0,-3)=9$ , $f(0,3)=9$ , $f\left(-\dfrac32,0\right)=\dfrac{729}4=182.25$ , and $f\left(\dfrac32,0\right)=\dfrac{729}4$ , so $f$ has a minimum value of 9 and a maximum value of 182.25.

ii. The Lagrangian is

$L(x,y,z,\lambda)=y^2-10z+\lambda(x^2+y^2+z^2-36)$

with critical points whenever

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We have $f(0,0,-6)=60$ , $f(0,0,6)=-60$ , $f(0,-\sqrt{11},-5)=61$ , and $f(0,\sqrt{11},-5)=61$ . So $f$ has a maximum value of 61 and a minimum value of -60.

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Answer:

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