Question 698·Medium·Algebra

Type $4x+2y=10$ in Desmos. Click its intercepts, where it crosses the axes: Desmos shows $(0,5)$ and $(2.5,0)$. These are two points on the given line, so you can use them to find its slope.

A slope is the change in $y$ divided by the change in $x$. From $(0,5)$ to $(2.5,0)$, type $\frac{0-5}{2.5-0}$. Desmos prints $-2$. The negative sign fits a line that goes down as $x$ increases.

Line $r$ is perpendicular, meaning it meets the given line at a right angle. Its slope must multiply $-2$ to give $-1$. Flip the fraction and change its sign: $-2=-\frac{2}{1}$ becomes $\frac12$, and $(-2)(\frac12)=-1$. Keeping $-2$ would make a line with the same direction, not a perpendicular one.

With slope $\frac12$, line $r$ has the form $y=\frac12x+b$, where $b$ is its $y$-intercept, the value of $y$ when $x=0$. Because the line passes through $(6,-5)$, put $6$ in for $x$ and $-5$ in for $y$: $-5=\frac12(6)+b$. Don't use $-5$ as $b$: the point's $x$-coordinate isn't $0$.

Type $-5\sim\frac12(6)+b$ to solve for the unknown intercept. Under PARAMETERS, Desmos shows $b=-8$. Put that into $y=\frac12x+b$: line $r$ is $y=\frac12x-8$. Choice D.