The ratio $PR:RQ = 1:2$ tells you how far along segment $PQ$ point $R$ is from $P$ to $Q$. If the whole segment has 3 equal parts, how many parts from $P$ to $R$?

Once you know that $R$ is a certain fraction of the way from $P$ to $Q$, write $R$ as a combination of $P$ and $Q$ (for example, $R = P + t(Q-P)$). Then plug those coordinates into $5x + 3y = 30$ and solve for $m$.

From the algebra steps, you should get an equation in $m$ of the form $-\dfrac{20}{m} + 6 = 30$. In Desmos, let the variable $x$ stand for $m$, and enter the two equations y = -20/x + 6 and y = 30.

In Desmos, tap the point where the graphs of y = -20/x + 6 and y = 30 intersect. The x-coordinate of this intersection is the value of $m$ that satisfies the condition in the problem.

Line ℓ has equation $y = mx + 6$.

• x-intercept $P$: On the x-axis, $y=0$.

So $P\left(-\dfrac{6}{m},\,0\right)$.

• y-intercept $Q$: On the y-axis, $x=0$.

So $Q(0,6)$.

The ratio $PR:RQ = 1:2$ means that from $P$ to $Q$, point $R$ is $\tfrac{1}{3}$ of the way along the segment (since $1+2=3$ total parts).

We can write

First find $Q - P$:

Then

So

Thus $R\left(-\dfrac{4}{m},\,2\right)$.

Because $R$ is on the line $5x + 3y = 30$, its coordinates must satisfy this equation.

Substitute $x = -\dfrac{4}{m}$ and $y = 2$ into $5x + 3y = 30$:

Simplify:

This is an equation in $m$ that we now solve.

We have

Subtract 6 from both sides:

Multiply both sides by $m$:

Now divide both sides by 24:

So $m = -\dfrac{5}{6}$, which corresponds to choice D.