Choice D is correct. The linear relationship between $x$ and $y$ can be represented by the equation $y = m x + b$ , where $m$ is the slope of the graph of this equation in the xy-plane and $b$ is the y-coordinate of the y-intercept. The slope of a line between any two points $left parenthesis x 1 comma y 1 right parenthesis$ and $left parenthesis x 2 comma y 2 right parenthesis$ on the line can be calculated using the slope formula $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ . Based on the table, the graph contains the points $(- 18, - 48)$ and $left parenthesis 7 comma 52 right parenthesis$ . Substituting $(- 18, - 48)$ and $left parenthesis 7 comma 52 right parenthesis$ for $left parenthesis x 1 comma y 1 right parenthesis$ and $left parenthesis x 2 comma y 2 right parenthesis$ , respectively, in the slope formula yields $m = \frac{52 - (- 48)}{7 - (- 18)}$ , which is equivalent to $m = \frac{100}{25}$ , or $m equals 4$ . Substituting $4$ for $m$ , $negative 18$ for $x$ , and $negative 48$ for $y$ in the equation $y = m x + b$ yields $- 48 = 4 (- 18) + b$ , or $- 48 = - 72 + b$ . Adding $72$ to both sides of this equation yields $24 equals b$ . Therefore, $m equals 4$ and $b equals 24$ . Substituting $4$ for $m$ and $24$ for $b$ in the equation $y = m x + b$ yields $y = 4 x + 24$ . Thus, the equation $y = 4 x + 24$ represents the linear relationship between $x$ and $y$ . It's also given that the graph of the linear equation representing this relationship in the xy-plane passes through the point $(\frac{1}{7}, a)$ . Substituting $\frac{1}{7}$ for $x$ and $a$ for $y$ in the equation $y = 4 x + 24$ yields $a = 4 (\frac{1}{7}) + 24$ , which is equivalent to $a = \frac{4}{7} + \frac{168}{7}$ , or $a = \frac{172}{7}$ .

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.