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3 years ago

−3x + 7y = −16 −9x + 5y = 16 slove this

Mathematics

1 answer:

Ahat [919]3 years ago

5 0

{-3x+7y=-16 ⇒ -3x=-16-7y (↓)

{-9x+5y=16

The second equation:

-9x+5y=16

3*(-3x)+5y=16 Substitute -3x (↑)

3* (-16-7y)+5y=16

-48-21y+5y=16 /+48

-16y=64 /:(-16)

y=-4

-3x=-16-7y

-3x=-16-7*(-4)=-16+28=12

-3x=12 /:(-3)

x=-4

Solution: (-4; -4)

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If $y=0$ , then $L_\lambda=0\implies x=\pm\dfrac32$ .
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$L(x,y,z,\lambda)=y^2-10z+\lambda(x^2+y^2+z^2-36)$

with critical points whenever

$L_x=2\lambda x=0\implies x=0$ (because we assume $\lambda\neq0$ )

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$L_z=-10+2\lambda z=0\implies z=\dfrac5\lambda$

$L_\lambda=x^2+y^2+z^2-36=0$

If $x=y=0$ , then $L_\lambda=0\implies z=\pm6$ .
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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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