Pick one labeled point, substitute its $x$- and $y$-values into an answer choice, and solve for $R$.

Since $R$ is a constant, substituting a second point should give the exact same value of $R$. If it does not, that choice cannot represent the line.

Enter the points as

• $A=(1,6)$
• $B=(9,1)$

so you can see whether a line passes through both.

Enter one choice, for example:

$5x + R y = 53$

Desmos will create a slider for $R$.

Move the slider and check whether the line goes through both plotted points $A$ and $B$ at the same time. If no slider value makes the line pass through both points, that choice cannot represent the graph.

Repeat for the other answer choices. The correct answer is the only equation for which some positive value of $R$ makes the line pass through both $A$ and $B$.

From the graph, the line passes through the labeled points $(1,6)$ and $(9,1)$.

Substitute $x=1$ and $y=6$.

• For $5x + Ry = 53$: $5(1) + 6R = 53 \Rightarrow 6R = 48 \Rightarrow R=8$.

• For $5x - Ry = 53$: $5(1) - 6R = 53 \Rightarrow -6R = 48 \Rightarrow R=-8$ (not positive).

• For $Rx + 6y = 53$: $R(1) + 6(6) = 53 \Rightarrow R+36=53 \Rightarrow R=17$.

• For $Rx - 6y = 53$: $R(1) - 6(6) = 53 \Rightarrow R-36=53 \Rightarrow R=89$.

Now substitute $x=9$ and $y=1$.

• For $5x + Ry = 53$, using $R=8$: $5(9)+8(1)=45+8=53$ (works).

• For $Rx + 6y = 53$, the $R$ needed would be $9R+6=53 \Rightarrow R=\frac{47}{9}$, which does not match $R=17$, so it cannot represent the line.

• For $Rx - 6y = 53$, the $R$ needed would be $9R-6=53 \Rightarrow R=\frac{59}{9}$, which does not match $R=89$, so it cannot represent the line.

Only $5x + Ry = 53$ gives one positive value of $R$ that fits both points on the line.

Therefore, the correct choice is $5x + Ry = 53$.