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Algebra / Linear equations in two variables Difficulty: Hard

$x$ $x$	$y$ $y$
$18$ $18$	$130$ $130$
$23$ $23$	$160$ $160$
$26$ $26$	$178$ $178$

For line $h$ , the table shows three values of $x$ and their corresponding values of $y$ . Line $k$ is the result of translating line $h$ down $5$ units in the xy-plane. What is the x-intercept of line $k$ ?

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Explanation

Choice D is correct. The equation of line $h$ can be written in slope-intercept form $y = m x + b$ , where $m$ is the slope of the line and $left parenthesis 0 comma b right parenthesis$ is the y-intercept of the line. It’s given that line $h$ contains the points $left parenthesis 18 comma 130 right parenthesis$ , $left parenthesis 23 comma 160 right parenthesis$ , and $left parenthesis 26 comma 178 right parenthesis$ . Therefore, its slope $m$ can be found as $\frac{160 - 130}{23 - 18}$ , or $6$ . Substituting $6$ for $m$ in the equation $y = m x + b$ yields $y = 6 x + b$ . Substituting $130$ for $y$ and $18$ for $x$ in this equation yields $130 = 6 (18) + b$ , or $130 = 108 + b$ . Subtracting $108$ from both sides of this equation yields $22 equals b$ . Substituting $22$ for $b$ in $y = 6 x + b$ yields $y = 6 x + 22$ . Since line $k$ is the result of translating line $h$ down $5$ units, an equation of line $k$ is $y = 6 x + 22 - 5$ , or $y = 6 x + 17$ . Substituting $0$ for $y$ in this equation yields $0 = 6 x + 17$ . Solving this equation for $x$ yields $x = - \frac{17}{6}$ . Therefore, the x-intercept of line $k$ is $(- \frac{17}{6}, 0)$ .

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.