Question 582·Hard·Algebra

The slope of $L$ is $3=\frac31$. Perpendicular slopes multiply to $-1$, so flip the fraction and change its sign: $M$ has slope $-\frac13$. Changing only the sign would give $-3$, which is not perpendicular to a line with slope $3$.

Use point-slope form, $y-y_1=m(x-x_1)$, which builds a line from its slope $m$ and a point $(x_1,y_1)$. With slope $3$ and the point on $L$ $(2,5)$, you get:

Line $M$ has slope $-\frac13$ and passes through $(-1,1)$, so point-slope form gives:

The plus sign comes from subtracting the point’s $x$-coordinate: $x-(-1)=x+1$.

Type both equations into Desmos and click their intersection, the point on both lines. Desmos labels it $(0.5,0.5)$. The second coordinate is $y$, and $0.5=\frac12$. So the $y$-coordinate of $P$ is $\frac12$. Choice C.