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Algebra / Linear equations in two variables Difficulty: Medium

In the xy-plane, line $k$ passes through the points $left parenthesis 0 comma negative 5 right parenthesis$ and $left parenthesis 1 comma negative 1 right parenthesis$ . Which equation defines line $k$ ?

$y = - x + \frac{1}{4}$

$y = \frac{1}{4} x - 5$

$y = - x + 4$

$y = 4 x - 5$

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Explanation

Choice D is correct. An equation defining a line in the xy-plane can be written in the form $y = m x + b$ , where $m$ represents the slope and $left parenthesis 0 comma b right parenthesis$ represents the y-intercept of the line. It’s given that line $k$ passes through the point $left parenthesis 0 comma negative 5 right parenthesis$ ; therefore, $b = - 5$ . The slope, $m$ , of a line can be found using any two points on the line, $left parenthesis x 1 comma y 1 right parenthesis$ and $left parenthesis x 2 comma y 2 right parenthesis$ , and the slope formula $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ . Substituting the points $left parenthesis 0 comma negative 5 right parenthesis$ and $left parenthesis 1 comma negative 1 right parenthesis$ for $left parenthesis x 1 comma y 1 right parenthesis$ and $left parenthesis x 2 comma y 2 right parenthesis$ , respectively, in the slope formula yields $m = \frac{(- 1 - (- 5))}{(1 - 0)}$ , or $m equals 4$ . Substituting $4$ for $m$ and $negative 5$ for $b$ in the equation $y = m x + b$ yields $y = 4 x - 5$ .

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.