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Algebra / Linear equations in two variables Difficulty: Medium

Line $p$ is defined by $2 y + 18 x = 9$ . Line $r$ is perpendicular to line $p$ in the xy-plane. What is the slope of line $r$ ?

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Explanation

Choice C is correct. It’s given that line $r$ is perpendicular to line $p$ in the xy-plane. This means that the slope of line $r$ is the negative reciprocal of the slope of line $p$ . If the equation for line $p$ is rewritten in slope-intercept form $y = m x + b$ , where $m$ and $b$ are constants, then $m$ is the slope of the line and $left parenthesis 0 comma b right parenthesis$ is its y-intercept. Subtracting $18 x$ from both sides of the equation $2 y + 18 x = 9$ yields $2 y = - 18 x + 9$ . Dividing both sides of this equation by $2$ yields $y = - 9 x + \frac{9}{2}$ . It follows that the slope of line $p$ is $negative 9$ . The negative reciprocal of a number is $negative 1$ divided by the number. Therefore, the negative reciprocal of $negative 9$ is $\frac{- 1}{- 9}$ , or $\frac{1}{9}$ . Thus, the slope of line $r$ is $\frac{1}{9}$ .

Choice A is incorrect. This is the slope of line $p$ , not line $r$ .

Choice B is incorrect. This is the reciprocal, not the negative reciprocal, of the slope of line $p$ .

Choice D is incorrect. This is the negative, not the negative reciprocal, of the slope of line $p$ .