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Algebra / Linear equations in one variable Difficulty: Medium

$open parenthesis, b minus 2, close parenthesis, times x, equals 8$

In the given equation, b is a constant. If the equation has no solution, what is the value of b ?

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Explanation

Choice A is correct. This equation has no solution when there is no value of x that produces a true statement. Solving the given equation for x by dividing both sides by $open parenthesis, b minus 2, close parenthesis$ gives $x equals, the fraction with numerator 8, and denominator, open parenthesis, b minus 2, close parenthesis$ . When $open parenthesis, b minus 2, close parenthesis, equals 0$ , the right-hand side of this equation will be undefined, and the equation will have no solution. Therefore, when $b equals 2$ , there is no value of x that satisfies the given equation.

Choices B, C, and D are incorrect. Substituting 4, 6, and 10 for b in the given equation yields exactly one solution, rather than no solution, for x. For example, substituting 4 for b in the given equation yields $open parenthesis, 4 minus 2, close parenthesis, times x, equals 8$ , or $2 x equals 8$ . Dividing both sides of $2 x equals 8$ by 2 yields $x equals 4$ . Similarly, if $b equals 6$ or $b equals 10$ , $x equals 2$ and $x equals 1$ , respectively.