Question 540921: If i represents the imaginary unit, what is the ordered pair of real numbers (x, y) for which x + xi + y – yi = -1 + 7i ?
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! x = 3
y = -4
here's why:
your equation is:
x + xi + y - yi = -1 + 7i
when you're dealing with complex numbers, the real parts are added together and the imaginary parts are added together.
this means tht:
x + y = -1
xi - yi = 7i
you can factor out the i in the second equation to get:
(x - y)i = 7i
if you divide both sides of this equation by i, you get:
x - y = 7
you now have 2 equations in x and y that need to be solved simultaneously.
they are:
x + y = -1
x - y = 7
if you add these 2 equations together you get:
2x = 6 (the + y and the - y cancel out).
if you divide both sides by 2 you get:
x = 3.
you can susbtitute in either of the 2 equations to get:
y = -4
subgstituting in the 2 equations, you get:
x + y becomes 3 + (-4) which is equal to 1, confirming the first equation in x and y is true.
x - y becomes 3 - (-4) which is equal to 7, confirming the second equation in x and y is true.
this confirms the values for x and y are good.
you can now subgstitute in your original equations to confirm that you will get the answer you desire.
the original equation is:
x + xi + y – yi = -1 + 7i
substitute 3 for x and -4 for y to get:
3 + 3i + (-4) - (-4i) = -1 + 7i
simplify the left side of the equation to get:
3 + 3i - 4 + 4i = -1 + 7i
simplify further to get:
-1 + 7i = -1 + 7i
this confirms the values for x and y are good.
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