Question 488·Medium·Algebra

Type $4x+3y=12$ in Desmos and click its axis crossings. Desmos shows $(0,4)$ and $(3,0)$. Slope is the change in $y$ divided by the change in $x$, so from the first point to the second, type $\frac{0-4}{3-0}$ in Desmos. It shows $-1.333\ldots$; tap the fraction button to see $-\frac43$.

Perpendicular lines meet at a right angle, and their slopes multiply to $-1$. The negative reciprocal of $-\frac43$ is $\frac34$: flip the fraction and change the sign. So line $m$ has slope $\frac34$, not $-\frac43$.

Write line $m$ in slope-intercept form, $y=\frac34x+b$, where $b$ is its value when $x=0$. The point $(6,-2)$ lies on $m$, so putting in its coordinates gives:

The point’s $-2$ is not the intercept because its $x$-coordinate is $6$, not $0$.

Subtract $\frac34(6)$ from both sides:

Type $-2-\frac34(6)$ in Desmos. It shows $-6.5$; tap the fraction button to see $-\frac{13}{2}$. That’s the $y$-intercept of line $m$. Choice D.