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Difficulty: Hard

The graph shows a linear relationship between $x$ and $y$ . Which equation represents this relationship, where $R$ is a positive constant?

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Explanation

Choice C is correct. The equation representing the linear relationship shown can be written in slope-intercept form $y = m x + b$ , where $m$ is the slope and $left parenthesis 0 comma b right parenthesis$ is the y-intercept of the line. The line shown passes through the points $left parenthesis 0 comma 6 right parenthesis$ and $left parenthesis 2 comma 0 right parenthesis$ . Given two points on a line, $left parenthesis x 1 comma y 1 right parenthesis$ and $left parenthesis x 2 comma y 2 right parenthesis$ , the slope of the line can be calculated using the equation $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ . Substituting $left parenthesis 0 comma 6 right parenthesis$ and $left parenthesis 2 comma 0 right parenthesis$ for $left parenthesis x 1 comma y 1 right parenthesis$ and $left parenthesis x 2 comma y 2 right parenthesis$ , respectively, in this equation yields $m = \frac{0 - 6}{2 - 0}$ , which is equivalent to $m = - \frac{6}{2}$ , or $m = - 3$ . Since $left parenthesis 0 comma 6 right parenthesis$ is the y-intercept, it follows that $b equals 6$ . Substituting $negative 3$ for $m$ and $6$ for $b$ in the equation $y = m x + b$ yields $y = - 3 x + 6$ . Adding $3 x$ to both sides of this equation yields $3 x + y = 6$ . Multiplying this equation by $6$ yields $18 x + 6 y = 36$ . It follows that the equation $18 x + R y = 36$ , where $R$ is a positive constant, represents this relationship.

Choice A is incorrect. The graph of this relationship passes through the point $left parenthesis 0 comma 2 right parenthesis$ , not $left parenthesis 0 comma 6 right parenthesis$ .

Choice B is incorrect. The graph of this relationship passes through the point $left parenthesis 0 comma 2 right parenthesis$ , not $left parenthesis 0 comma 6 right parenthesis$ .

Choice D is incorrect. The graph of this relationship passes through the point $left parenthesis negative 2 comma 0 right parenthesis$ , not $left parenthesis 2 comma 0 right parenthesis$ .