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2 years ago

5x+2y=-4 -x+2y=8 Solve the system

Mathematics

1 answer:

maria [59]2 years ago

6 0

Subtract the second equation from the first.
.. (5x +2y) -(-x +2y) = (-4) -(8)
.. 6x = -12
.. x = -2

.. -(-2) +2y = 8
.. 2y = 6
.. y = 3

The solution is (x, y) = (-2, 3).

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$A = \int\limits^3__-3}{9}-{x^{2}} \, dx = 36$

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The equations are:

$y = x^{2} + 2x + 3$

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The two graphs intersect when:

$x^{2} + 2x + 3 = 2x + 12$

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To find the area under the curve for the first equation:

$A_{1} = \int\limits^3__-3}{x^{2} + 2x + 3} \, dx$

To find the area under the curve for the second equation:

$A_{2} = \int\limits^3__-3}{2x + 12} \, dx$

To find the total area:

$A = A_{2} -A_{1} = \int\limits^3__-3}{2x + 12} \, dx -\int\limits^3__-3}{x^{2} + 2x + 3} \, dx$

Simplifying the equation:

$A = \int\limits^3__-3}{2x + 12}-({x^{2} + 2x + 3}) \, dx = \int\limits^3__-3}{9}-{x^{2}} \, dx$

Note: The reason the area is equal to the area two minus area one is that the line, area 2, is above the region of interest (see image).

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