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Algebra / Linear equations in two variables Difficulty: Hard

In the xy-plane, line $l$ passes through the point $left parenthesis 0 comma 0 right parenthesis$ and is parallel to the line represented by the equation $y = 8 x + 2$ . If line $l$ also passes through the point $left parenthesis 3 comma d right parenthesis$ , what is the value of $d$ ?

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Explanation

The correct answer is $24$ . A line in the xy-plane can be defined by the equation $y = m x + b$ , where $m$ is the slope of the line and $b$ is the y-coordinate of the y-intercept of the line. It's given that line $l$ passes through the point $left parenthesis 0 comma 0 right parenthesis$ . Therefore, the y-coordinate of the y-intercept of line $l$ is $0$ . It's given that line $l$ is parallel to the line represented by the equation $y = 8 x + 2$ . Since parallel lines have the same slope, it follows that the slope of line $l$ is $8$ . Therefore, line $l$ can be defined by an equation in the form $y = m x + b$ , where $m equals 8$ and $b equals 0$ . Substituting $8$ for $m$ and $0$ for $b$ in $y = m x + b$ yields the equation $y = 8 x + 0$ , or $y equals 8 x$ . If line $l$ passes through the point $left parenthesis 3 comma d right parenthesis$ , then when $x equals 3$ , $y = d$ for the equation $y equals 8 x$ . Substituting $3$ for $x$ and $d$ for $y$ in the equation $y equals 8 x$ yields $d equals 8 left parenthesis 3 right parenthesis$ , or $d equals 24$ .