Question 283·Hard·Algebra

At an intercept, one coordinate is zero. Since $Q$ lies on the $y$-axis, put $x=0$ in $y=mx+6$ to get $Q=(0,6)$. Since $P$ lies on the $x$-axis, its height is $0$. You don’t need its $x$-coordinate.

The segment ratio $PR:RQ=1:2$ puts $R$ one of three equal parts from $P$. The same fraction of a segment gives the same fraction of its change in height. From $P$ to $Q$, the height rises from $0$ to $6$, so $R$’s height is $\frac{6}{3}=2$. Using two parts would put $R$ nearer $Q$, not $P$.

Type $5x+3y=30$ and $y=2$ in Desmos. Their intersection is on the given second line at $R$’s height. Click it: Desmos shows $(4.8,2)$, so $R$’s $x$-coordinate is $4.8$.

$R$ also lies on line $\ell$, so put its coordinates into $y=mx+6$: $2=4.8m+6$. This uses the line whose slope is $m$, not the second line that helped locate $R$.

Type $2\sim4.8m+6$ in Desmos. The $\sim$ tells Desmos to find the parameter $m$ rather than make a slider. Under PARAMETERS, it shows $m=-0.833333\ldots$. Type $m$ on its own line and tap the fraction button to see $-\frac{5}{6}$. So line $\ell$ has slope $-\frac{5}{6}$. Choice D.