Question 494599: Please hep me solve this equation: (a+bi)+18(a-bi)=7+1i a=? b=?
Answer by bucky(2189)   (Show Source):
You can put this solution on YOUR website! Given:
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(a+bi)+18(a-bi)=7+1i
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Simplify the left side of this equation. Begin by doing the distributed multiplication. To do this multiplication, multiply the 18 times each of the terms in the associated set of parentheses. So multiply 18 times "a" and also 18 times -bi. When you do that the left side changes and the equation becomes:
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(a + bi) + 18a - 18bi = 7 + 1i
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Then remove the remaining set of parentheses. Since this set of parentheses is not preceded by a minus sign, do not change the signs of the terms inside. The left side of the equation becomes as shown:
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a + bi + 18a - 18bi = 7 + 1i
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Collect the like terms on the left side to get:
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a + 18a + bi - 18bi = 7 + 1i
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The two terms that containing "a" are the real terms because they do not contain an "i". When added together these two real terms (a and +18a) total +19a.
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And the two terms containing "i" are the imaginary terms. These terms are +bi and -18bi. When they are totaled they give -17bi. This reduces the equation to:
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19a - 17bi = 7 + 1i
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Now look at the real part of the left side. It is 19a and it must equal the real part of the right side which is 7. So we can say:
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19a = 7
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Solve for "a" by dividing both sides of this equation by 19 and you get:
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a = 7/19
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And now look at the imaginary part of the left side. The multiplier of the "i" on the left side is -17b. It must equal the multiplier of the "i" on the right side which is +1. So we can write the equation:
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-17b = +1
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Solve for b by dividing both sides by -17 to get:
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b = 1/(-17) or b = -(1/17)
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You now know that a = 7/19 and b = -(1/17) and those are the answers to this problem.
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Hope this helps you to understand how you can compare the real terms of both sides of an equation and then compare the imaginary terms of both sides.
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