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Alla [95]
3 years ago
11

Show how to use substitution or elimination to solve the following system of equations. Write the solution as an ordered pair.

Mathematics
1 answer:
max2010maxim [7]3 years ago
3 0

Answer:

  (4, 6)

Step-by-step explanation:

<u>Substitution</u>

Using the second equation, we can substitute for y in the first equation.

  4x -(2x-2) = 10

  2x +2 = 10 . . . . simplify

  x +1 = 5 . . . . . . .divide by 2

  x = 4 . . . . . . . . . subtract 1

  y = 2(4) -2) = 6 . . . . substitute for x in the equation for y

The solution is (x, y) = (4, 6).

__

<u>Elimination</u>

Often, we like to start with both equations in standard form when we solve by elimination. That is nice, but not completely necessary.

We can eliminate the y-variable by simply adding the two given equations.

  (4x -y) +(y) = (10) +(2x -2)

  4x = 8 + 2x . . . . . collect terms. The y-variable has been eliminated.

  2x = 8 . . . . . . . . . subtract 2x

  x = 4 . . . . . . . . . . divide by 2

Y can be found using the second equation.

  (x, y) = (4, 6)

__

We could have eliminated the x-variable by multiplying the second equation by 2, then adding the two equations.

  (4x -y) +2(y) = (10) +2(2x -2)

  y = 6 . . . . . . . . . subtract 4x from both sides. The x-variable has been eliminated.

Now, x can be found using either equation.

  6 = 2x -2 . . . substitute for y in the second equation

  3 = x - 1 . . . divide by 2

  x = 4 . . . . . add 1

  (x, y) = (4, 6)

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Answer: y=Ce^(^3^t^{^9}^)

Step-by-step explanation:

Beginning with the first differential equation:

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Integrate both sides. Both sides will produce an integration constant, but I will merge them together into a single integration constant on the right side:

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We can take out the 'C' of the exponential using a rule of exponents. Addition in an exponent can be broken up into a product of their bases:

y=e^(^3^t^{^9}^)e^C

The term e^C is just another constant, so with impunity, I can absorb everything into a single constant:

y=Ce^(^3^t^{^9}^)

To check the answer by differentiation, you require the chain rule. Differentiating an exponential gives back the exponential, but you must multiply by the derivative of the inside. We get:

\frac{d}{dx} (y)=\frac{d}{dx}(Ce^(^3^t^{^9}^))

\frac{dy}{dx} =(Ce^(^3^t^{^9}^))*\frac{d}{dx}(3t^9)

\frac{dy}{dx} =(Ce^(^3^t^{^9}^))*27t^8

Now check if the derivative equals the right side of the original differential equation:

(Ce^(^3^t^{^9}^))*27t^8=27t^8*y(t)

Ce^(^3^t^{^9}^)*27t^8=27t^8*Ce^(^3^t^{^9}^)

QED

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