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Solve system of equations and examine how many solutions have? x+i*y=-1 2x+3y=3i

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greenbel | (Level 2) Honors

Posted August 30, 2013 at 7:50 AM via web

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Solve system of equations and examine how many solutions have?

x+i*y=-1

2x+3y=3i

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sciencesolve | Teacher | (Level 3) Educator Emeritus

Posted August 30, 2013 at 8:29 AM (Answer #1)

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You may use substitution method, hence, you may use the first equation to express x in terms of y, such that:

`x = -1 - i*y`

Replacing `-1 - i*y` for `x` in the bottom equation, yields:

`2(-1 - i*y) + 3y = 3i`

`-2 - 2i*y + 3y = 3i`

You need to keep the terms that contain y to the left side, such that:

`y(-2i + 3) = 3i + 2 => y = (3i + 2)/(3 - 2i)`

`y = ((3i + 2)(3 + 2i))/((3 - 2i)(3 + 2i))`

`y = ((3i + 2)(3 + 2i))/(9 - 4i^2)`

Replacing `-1` for `i^2` yields:

`y = ((3i + 2)(3 + 2i))/13 => y = (9i + 6i^2 + 6 + 4i)/13`

`y = (9i -6 + 6 + 4i)/13`

`y = (13i)/13 => y = i`

Replacing `i` for `y` in equation `x = -1 - i*y` yields:

`x = -1 - i*i => x = -1 - i^2 => x = -1 + 1 => x = 0`

Hence, evaluating the solution to the system of equations, yields that the system has the unique solution `x = 0, y = i.`

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