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Difficulty: Medium

$x$ $x$	$y$ $y$
$- 6$ $negative 6$	$65$ $65$
$- 3$ $negative 3$	$56$ $56$
$3$ $3$	$38$ $38$
$6$ $6$	$29$ $29$

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

$9 x + 3 y = 141$

$9 x + 3 y = 3$

$3 x + 9 y = 141$

$3 x + 9 y = 3$

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Explanation

Choice A is correct. An equation representing the linear relationship between $x$ $x$ and $y$ $y$ can be written in slope-intercept form $y = m x + b$ $y = m x + b$ , where $m$ $m$ is the slope of the graph of the equation in the xy-plane and $(0, b)$ $left parenthesis 0 comma b right parenthesis$ is the y-intercept. The slope, $m$ $m$ , can be calculated using two ordered pairs, $(x_{1}, y_{1})$ $left parenthesis x 1 comma y 1 right parenthesis$ and $(x_{2}, y_{2})$ $left parenthesis x 2 comma y 2 right parenthesis$ , and the formula $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ . Substituting the ordered pairs $(- 6, 65)$ $left parenthesis negative 6 comma 65 right parenthesis$ and $(6, 29)$ $left parenthesis 6 comma 29 right parenthesis$ from the table for $(x_{1}, y_{1})$ $left parenthesis x 1 comma y 1 right parenthesis$ and $left parenthesis x 2 comma y 2 right parenthesis$ , respectively, in this formula yields $m = \frac{29 - 65}{6 - (- 6)}$ , which is equivalent to $m = \frac{- 36}{12}$ , or $m = - 3$ . Substituting $negative 3$ for $m$ in the formula $y = m x + b$ yields $y = - 3 x + b$ . Substituting the point $left parenthesis negative 6 comma 65 right parenthesis$ into this equation yields $65 = - 3 (- 6) + b$ , or $65 = 18 + b$ . Subtracting $18$ from both sides of this equation yields $47 equals b$ . Substituting $47$ for $b$ in the equation $y = - 3 x + b$ yields $y = - 3 x + 47$ . Adding $3 x$ to both sides of this equation yields $3 x + y = 47$ . Multiplying both sides of this equation by $3$ yields $9 x + 3 y = 141$ .

Choice B is incorrect. Substituting the point $left parenthesis negative 6 comma 65 right parenthesis$ from the table into this equation yields $9 (- 6) + 3 (65) = 3$ , or $141 equals 3$ , which is false.

Choice C is incorrect. Substituting the point $left parenthesis negative 6 comma 65 right parenthesis$ from the table into this equation yields $3 (- 6) + 9 (65) = 141$ , or $567 equals 141$ , which is false.

Choice D is incorrect. Substituting the point $left parenthesis negative 6 comma 65 right parenthesis$ from the table into this equation yields $3 (- 6) + 9 (65) = 3$ , or $567 equals 3$ , which is false.