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Algebra / Linear equations in two variables Difficulty: Hard

$x$ $x$	$y$ $y$
$- 2 s$ $minus 2 s$	$24$ $24$
$- s$ $- s$	$21$ $21$
$s$ $s$	$15$ $15$

The table shows three values of $x$ and their corresponding values of $y$ , where $s$ is a constant. There is a linear relationship between $x$ and $y$ . Which of the following equations represents this relationship?

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Explanation

Choice B is correct. The linear relationship between $x$ and $y$ can be represented by an equation of the form $y - y_{1} = m (x - x_{1})$ , where $m$ is the slope of the graph of the equation in the xy-plane and $left parenthesis x 1 comma y 1 right parenthesis$ is a point on the graph. The slope of a line can be found using two points on the line and the slope formula $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ . Each value of $x$ and its corresponding value of $y$ in the table can be represented by a point $(x, y)$ . Substituting the points $left parenthesis negative s comma 21 right parenthesis$ and $left parenthesis s comma 15 right parenthesis$ 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 $m = \frac{15 - 21}{s - (- s)}$ , which gives $m = \frac{- 6}{2 s}$ , or $m = - \frac{3}{s}$ . Substituting $- \frac{3}{s}$ for $m$ and the point $left parenthesis s comma 15 right parenthesis$ for $left parenthesis x 1 comma y 1 right parenthesis$ in the equation $y - y_{1} = m (x - x_{1})$ yields $y - 15 = - \frac{3}{s} (x - s)$ . Distributing $- \frac{3}{s}$ on the right-hand side of this equation yields $y - 15 = - \frac{3 x}{s} + 3$ . Adding $15$ to each side of this equation yields $y = - \frac{3 x}{s} + 18$ . Multiplying each side of this equation by $s$ yields $s y = - 3 x + 18 s$ . Adding $3 x$ to each side of this equation yields $3 x + s y = 18 s$ . Therefore, the equation $3 x + s y = 18 s$ represents this relationship.

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

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

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