Question 118051This question is from textbook Advanced Mathematics - precalculus with discrete mathamatics and data analysis
: Find real numbers of x and y such that :
(2x+y) + (3-5x)i = 1-7i
This question is from textbook Advanced Mathematics - precalculus with discrete mathamatics and data analysis
Answer by bucky(2189)   (Show Source):
You can put this solution on YOUR website! This problem can be solved by recognizing that both the left side and the right side have real
and imaginary parts that are separate.
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The real part of the left side is 2x + y and the real part of the right side is 1. These two must
be equal. So one of the equations that can be written is:
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2x + y = 1.
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Next, look at the multipliers of i on both the left and right sides. On the left side the multiplier
of i is 3 - 5x. On the right side the multiplier of i is -7. These two multipliers must equal
to make the imaginary parts equal. So another equation that can be written is:
.
3 - 5x = -7
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Subtract 3 from both sides to get rid of the 3 on the left side and the resulting
equation is:
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-5x = -10
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Solve for x by dividing both sides by -5 and you have:
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x = -10/-5 = +2
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Now that x has been found, return to the first equation and substitute +2 for x and solve
for y. The first equation was:
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2x + y = 1
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Substitute +2 for x and this equation becomes:
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2(2) + y = 1
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Multiply out the first term on the left side to get:
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4 + y = 1
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Get rid of the 4 on the left side by subtracting 4 from both sides. This reduces the equation
to:
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y = 1 - 4 = -3
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So the answers to this problem are: x = +2, y = -3.
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Check by substituting these two values into the original problem:
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(2x+y) + (3-5x)i = 1-7i
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Substitute the values that were found to get:
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(2(2)+ (-3)) + (3 -5(2))i = 1 - 7i
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(4 - 3) + (3 - 10)i = 1 - 7i
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1 - 7i = 1 - 7i
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Checks out, so the answers are correct: ... x = +2, y = -3.
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Hope this helps you to understand the problem. The basic lesson is that the real parts on
both sides of the equal sign must be equal, and the imaginary parts on both sides of the
equal sign must also be equal.
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