Choice A is correct. An equation representing the linear relationship between $x$ and $y$ can be written in slope-intercept form $y = m x + b$ , where $m$ is the slope of the graph of the equation in the xy-plane and $left parenthesis 0 comma b right parenthesis$ is the y-intercept. The slope, $m$ , can be calculated using two ordered pairs, $left parenthesis x 1 comma y 1 right parenthesis$ and $left parenthesis x 2 comma y 2 right parenthesis$ , and the formula $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ . Substituting the ordered pairs $left parenthesis negative 6 comma 65 right parenthesis$ and $left parenthesis 6 comma 29 right parenthesis$ from the table for $left parenthesis x 1 comma y 1 right parenthesis$ 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.