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Algebra / Linear equations in one variable Difficulty: Hard

The equation $9 x plus 5, equals a, times, open parenthesis, x plus b, close parenthesis$ , where a and b are constants, has no solutions. Which of the following must be true?

I. $a, equals 9$

II. $b equals 5$

III. $b is not equal to five ninths$

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Explanation

Choice D is correct. For a linear equation in a form $a, x plus b, equals, c x plus d$ to have no solutions, the x-terms must have equal coefficients and the remaining terms must not be equal. Expanding the right-hand side of the given equation yields $9 x plus 5, equals, a, x plus a, b$ . Inspecting the x-terms, 9 must equal a, so statement I must be true. Inspecting the remaining terms, 5 can’t equal $9 b$ . Dividing both of these quantities by 9 yields that b can’t equal $five ninths$ . Therefore, statement III must be true. Since b can have any value other than $five ninths$ , statement II may or may not be true.

Choice A is incorrect. For the given equation to have no solution, both $a, equals 9$ and $b is not equal to five ninths$ must be true. Choice B is incorrect because it must also be true that $b is not equal to five ninths$ . Choice C is incorrect because when $a, equals 9$ , there are many values of b that lead to an equation having no solution. That is, b might be 5, but b isn’t required to be 5.