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Question 1063361: My book doesn't show me how to do this. Any steps and explanations would be great.
Assume that a,b, and c are integers and a is not equal to 0.
a. Proof Prove that the solution of the linear equation ax-b=c must be a rational number.
b. Writing Describe the values of a, b, and c for which the solutions of ax^2+b=c are rational.
Answer by rothauserc(4718)   (Show Source):
You can put this solution on YOUR website! The set of rational numbers with the operations of addition, subtraction, multiplication and division defined, this structure is called a Field(F)
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We are given a, b, c are integers and a is not equal to 0, therefore a, b, c are rational numbers
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Note that a = a / 1, b = b / 1 and c = c / 1
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a. ax - b = c
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add b to both sides of =, b + (-b) = 0 since -b is the additive inverse of b
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ax = c + b
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since a is not = 0 there exists a^(-1) which is the multiplicative inverse of a, that is a * a^(-1) = 1 where * is multiplication
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x = (c + b) * a^(-1)
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a^(-1) = 1/a
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x = (c + b) / a
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Note that F is closed under addition, this means that given c and b are rational, then c + b = d is rational
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then d / a is rational(can be expressed as a fraction)
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b. ax^2 + b = c
ax^2 = (c - b)
x^2 = (c - b) / a
x = sqrt( (c - b) / a )
x is rational if a not = 0, b , c are rational numbers with c > or = b
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Note that 0 is a rational number
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