Choice $A$ is correct. Each of the choices is a linear equation in the form $y=mx+b$, where $m$ and $b$ are constants. In this equation, $m$ represents the change in $y$ for each increase in $x$ by $1$. From the table, it can be determined that the value of $y$ increases by $2$ for each increase in $x$ by $1$. In other words, for the pairs of $x$ and $y$ in the given table, $m=2$. The value of $b$ can be found by substituting the values of $x$ and $y$ from any row of the table and substituting the value of $m$ into the equation $y=mx+b$ and then solving for $b$. For example, using $x=1$, $y=5$, and $m=2$ yields $5=2(1)+b$. Solving for $b$ yields $b=3$. Therefore, the equation $y=2x+3$ could represent the relationship between $x$ and $y$ in the given table.

Alternatively, if an equation represents the relationship between $x$ and $y$, then when each pair of $x$ and $y$ from the table is substituted into the equation, the result will be a true statement. Of the equations given, the equation $y=2x+3$ in choice $A$ is the only equation that results in a true statement when each of the pairs of $x$ and $y$ are substituted into the equation.

Choices $B$ , $C$, and $D$ are incorrect because when at least one pair of $x$ and $y$  from the table is substituted into the equations given in these choices, the result is a false statement. For example, when the pair $x=4$ and $y=11$ is substituted into the equation in choice $B$, the result is $11=3(4)-2$, or $11=10$, which is false.