The relationship between two variables, x and y , is linear. For every increase in the

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 $y=mx+b$, 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 $y=mx+b$ yields $y=8x+b$. 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 $y=8x+b$ yields $18=8\left(2\right)+b$, or $18=16+b$. Subtracting $16$ from each side of this equation yields $2=b$. Substituting $2$ for $b$ in the equation $y=8x+b$ yields $y=8x+2$. Therefore, the equation $y=8x+2$ 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 $18$.