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Algebra / Linear equations in two variables Difficulty: Medium

In the xy-plane, line $s$ passes through the point $left parenthesis 0 comma 0 right parenthesis$ and is parallel to the line represented by the equation $y = 18 x + 2$ . If line $s$ also passes through the point $left parenthesis 4 comma d right parenthesis$ , what is the value of $d$ ?

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Explanation

Choice C is correct. A line in the xy-plane can be represented by an equation of the form $y = m x + b$ , where $m$ is the slope and $b$ is the y-coordinate of the y-intercept of the line. It's given that line $s$ passes through the point $left parenthesis 0 comma 0 right parenthesis$ . Therefore, the y-coordinate of the y-intercept of line $s$ is $0$ . It's also given that line $s$ is parallel to the line represented by the equation $y = 18 x + 2$ . Since parallel lines have the same slope, it follows that the slope of line $s$ is $18$ . Therefore, line $s$ can be represented by the equation $y = m x + b$ , where $m equals 18$ and $b equals 0$ . Substituting $18$ for $m$ and $0$ for $b$ in $y = m x + b$ yields the equation $y = 18 x + 0$ , or $y equals 18 x$ . If line $s$ passes through the point $left parenthesis 4 comma d right parenthesis$ , then when $x equals 4$ , $y = d$ for the equation $y equals 18 x$ . Substituting $4$ for $x$ and $d$ for $y$ in this equation yields $d equals 18 left parenthesis 4 right parenthesis$ , or $d equals 72$ .

Choice A is incorrect. This is the y-coordinate of the y-intercept of the line represented by the equation $y = 18 x + 2$ .

Choice B is incorrect. This is the slope of the line represented by the equation $y = 18 x + 2$ .

Choice D is incorrect. The line represented by the equation $y = 18 x + 2$ , not line $s$ , passes through the point $left parenthesis 4 comma 74 right parenthesis$ .