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Algebra / Linear equations in two variables Difficulty: Hard

The graph of $7 x + 2 y = - 31$ in the $x y$ -plane has an $x$ -intercept at $left parenthesis a comma 0 right parenthesis$ and a $y$ -intercept at $left parenthesis 0 comma b right parenthesis$ , where $a$ and $b$ are constants. What is the value of $\frac{b}{a}$ ?

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Explanation

Choice D is correct. The x-coordinate $a$ of the x-intercept $left parenthesis a comma 0 right parenthesis$ can be found by substituting $0$ for $y$ in the given equation, which gives $7 x + 2 (0) = - 31$ , or $7 x = - 31$ . Dividing both sides of this equation by $7$ yields $x = - \frac{31}{7}$ . Therefore, the value of $a$ is $- \frac{31}{7}$ . The y-coordinate $b$ of the y-intercept $left parenthesis 0 comma b right parenthesis$ can be found by substituting $0$ for $x$ in the given equation, which gives $7 (0) + 2 y = - 31$ , or $2 y = - 31$ . Dividing both sides of this equation by $2$ yields $y = - \frac{31}{2}$ . Therefore, the value of $b$ is $- \frac{31}{2}$ . It follows that the value of $\frac{b}{a}$ is $\frac{- \frac{31}{2}}{- \frac{31}{7}}$ , which is equivalent to $left parenthesis StartFraction 31 Over 2 EndFraction right parenthesis left parenthesis seven thirty firsts right parenthesis$ , or $\frac{7}{2}$ .

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.