Choice C is correct. It’s given that the relationship between $x$ and $y$ is linear. An equation representing a linear relationship can be written in the form $y = m x + b$ , where $m$ is the slope and $b$ is the y-coordinate of the y-intercept of the graph of the relationship in the xy-plane. It’s given that for every increase in the value of $x$ by $1$ , the value of $y$ increases by $8$ . The slope of a line can be expressed as the change in $y$ over the change in $x$ . Thus, the slope, $m$ , of the line representing this relationship can be expressed as $\frac{8}{1}$ , or $8$ . Substituting $8$ for $m$ in the equation $y = m x + b$ yields $y = 8 x + b$ . It's also given that when the value of $x$ is $2$ , the value of $y$ is $18$ . Substituting $2$ for $x$ and $18$ for $y$ in the equation $y = 8 x + b$ yields $18 = 8 (2) + b$ , or $18 = 16 + b$ . Subtracting $16$ from each side of this equation yields $2 equals b$ . Substituting $2$ for $b$ in the equation $y = 8 x + b$ yields $y = 8 x + 2$ . Therefore, the equation $y = 8 x + 2$ represents this relationship.

Choice A is incorrect. This equation represents a relationship where for every increase in the value of $x$ by $1$ , the value of $y$ increases by $2$ , not $8$ , and when the value of $x$ is $2$ , the value of $y$ is $22$ , not $18$ .

Choice B is incorrect. This equation represents a relationship where for every increase in the value of $x$ by $1$ , the value of $y$ increases by $2$ , not $8$ , and when the value of $x$ is $2$ , the value of $y$ is $12$ , not $18$ .

Choice D is incorrect. This equation represents a relationship where for every increase in the value of $x$ by $1$ , the value of $y$ increases by $3$ , not $8$ , and when the value of $x$ is $2$ , the value of $y$ is $32$ , not $18$ .