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Algebra / Linear equations in two variables Difficulty: Medium

Line $t$ in the xy-plane has a slope of $- \frac{1}{3}$ and passes through the point $left parenthesis 9 comma 10 right parenthesis$ . Which equation defines line $t$ ?

$y = 13 x - \frac{1}{3}$

$y = 9 x + 10$

$y = - \frac{x}{3} + 10$

$y = - \frac{x}{3} + 13$

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Explanation

Choice D is correct. The equation that defines line $t$ in the xy-plane can be written in slope-intercept form $y = m x + b$ , where $m$ is the slope of line $t$ and $left parenthesis 0 comma b right parenthesis$ is its y-intercept. It’s given that line $t$ has a slope of $- \frac{1}{3}$ . Therefore, $m = - \frac{1}{3}$ . Substituting $- \frac{1}{3}$ for $m$ in the equation $y = m x + b$ yields $y = - \frac{1}{3} x + b$ , or $y = - \frac{x}{3} + b$ . It’s also given that line $t$ passes through the point $left parenthesis 9 comma 10 right parenthesis$ . Substituting $9$ for $x$ and $10$ for $y$ in the equation $y = - \frac{x}{3} + b$ yields $10 = - \frac{9}{3} + b$ , or $10 = - 3 + b$ . Adding $3$ to both sides of this equation yields $13 equals b$ . Substituting $13$ for $b$ in the equation $y = - \frac{x}{3} + b$ yields $y = - \frac{x}{3} + 13$ .

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

Choice B is incorrect. This equation defines a line that has a slope of $9$ , not $- \frac{1}{3}$ , and passes through the point $left parenthesis 0 comma 10 right parenthesis$ , not $left parenthesis 9 comma 10 right parenthesis$ .

Choice C is incorrect. This equation defines a line that passes through the point $left parenthesis 0 comma 10 right parenthesis$ , not $left parenthesis 9 comma 10 right parenthesis$ .