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3 years ago

1. 3x + x - 2x + 8 = 3x + x

Mathematics

1 answer:

FinnZ [79.3K]3 years ago

3 0

Step-by-step explanation:

1. 3x+x-2x+8=3x+x

x-2x+8=x

x+8=x+2x

8=2x

8÷2=4

x=4

2. 2(2x+4)-2x=x+18

8+4x-2x=x+18

8+2x=x+18

2x=x+10

x=10

3. 2(x+3)+3x

2(5+3)+15

16+15

4. 3(2x)-3x+4=2x+5

6x-3x+4=2x+5

3x+4=2x+5

3x-2x+4=5

x=1

5. 3x+2x+x=x+6

6x=x+6

5x=6

x=6/5

6. 2(x+4)+x=16

2x+8+x=16

3x+8=16

3x=8

x=8/3

Hope I calculated it right you can check it in calculator

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We want to find $f$ such that $\nabla f=\vec F$ . This means

$\dfrac{\partial f}{\partial x}=-6x+5y\implies f(x,y)=-3x^2+5xy+g(y)$

$\dfrac{\partial f}{\partial y}=5x+10y=5x+\dfrac{\mathrm dg}{\mathrm dy}\implies\dfrac{\mathrm dg}{\mathrm dy}=10y\implies g(y)=5y^2+C$

$\implies\boxed{f(x,y)=-3x^2+5xy+5y^2+C}$

so $\vec F$ is conservative.

$\mathrm{curl}\vec F=\left(\dfrac{\partial(-2y)}{\partial z}-\dfrac{\partial(1)}{\partial y}\right)\vec\imath+\left(\dfrac{\partial(-3x)}{\partial z}-\dfrac{\partial(1)}{\partial z}\right)\vec\jmath+\left(\dfrac{\partial(-2y)}{\partial x}-\dfrac{\partial(-3x)}{\partial y}\right)\vec k=\vec0$

Then

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$\dfrac{\partial f}{\partial y}=-2y=\dfrac{\partial g}{\partial y}\implies g(y,z)=-y^2+h(y)$

$\dfrac{\partial f}{\partial z}=1=\dfrac{\mathrm dh}{\mathrm dz}\implies h(z)=z+C$

$\implies\boxed{f(x,y,z)=-\dfrac32x^2-y^2+z+C}$

so $\vec F$ is conservative.

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so $\vec F$ is not conservative.

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