Mathematics: Solve the following system of equations by the substitution method. 8x = 2y + 5 3x = y + 7 what is the solution set? {(-41/2, -9/2)} {(-9/2, -41/2)} ø

By the chain rule, if , thenWe have , for which• • and and , so that• • Putting everything together, we get...

Solve the following system of equations by the substitution

Ответ:

jackieespi1461

17.01.2022

8x = 2y + 5
3x = y + 7

We will use the second equation to substitute in the first one. Here's how to do it:
3x = y + 7
y = 3x - 7 (now substitute this in the first equation)

8x = 2y + 5
8x = 2(3x - 7) + 5
8x = 6x - 14 + 5
8x = 6x - 9
6x - 9 = 8x
6x - 8x = 9
-2x = 9
x = -9/2 (this is the first coordinate, now we need to find y)

We use the second equation again:
3x = y + 7
y + 7 = 3x
y + 7 = 3(-9/2)
y + 7 = -27/2
y = -27/2 - 7
y = -27/2 - 14/2
y = -41/2 (this is the second coordinate)

This means that the solution set is the second option: {(-9/2, -41/2)}

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Ответ:

kaleahlove13

16.03.2023

Mathematics

By the chain rule, if z=f(x,y)=f(x(t),y(t)) , then

$\dfrac{\mathrm dz}{\mathrm dt} = \dfrac{\partial f}{\partial x}\dfrac{\mathrm dx}{\mathrm dt} + \dfrac{\partial f}{\partial y}\dfrac{\mathrm dy}{\mathrm dt}$

We have f(x,y)=(x+y)e^y , for which

• $\dfrac{\partial f}{\partial x} = e^y$

• $\dfrac{\partial f}{\partial y} = e^y + (x+y)e^y = (x+y+1)e^y$

and x(t)=5t and y(t)=1-t^2 , so that

• $\dfrac{\mathrm dx}{\mathrm dt} = 5$

• $\dfrac{\mathrm dy}{\mathrm dt} = -2t$

Putting everything together, we get

$\dfrac{\mathrm dz}{\mathrm dt} = 5e^y - 2t(x+y+1)e^y \\\\ \dfrac{\mathrm dz}{\mathrm dt} = 5e^{1-t^2} - 2t(5t-t^2+2)e^{1-t^2} = \boxed{\left(2t^3-10t^2-4t+5\right)e^{1-t^2}}$

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Solve the following system of equations by the substitution method.

8x = 2y + 5

3x = y + 7

what is the solution set?

{(-41/2, -9/2)}
{(-9/2, -41/2)}
ø

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