Question 412·Medium·Algebra

An intercept is where a line crosses an axis. Type $4x-2y=6$ in Desmos and click its axis crossings. Desmos shows $(0,-3)$ and $(1.5,0)$, two points you can use to find the line’s slope.

The slope is the change in $y$ divided by the change in $x$. Going from $(0,-3)$ to $(1.5,0)$, type $\frac{0-(-3)}{1.5-0}$. Desmos prints $2$, so the given line rises $2$ units for every $1$ unit to the right.

Parallel lines have equal slopes, so the new line has slope $2$ too. Write it as $y=2x+b$, where $b$ is the $y$-intercept, the value of $y$ when $x=0$. Parallel sets the slope; the point sets the intercept.

Because the line passes through $(3,-2)$, $x=3$ must give $y=-2$. Put both coordinates into $y=2x+b$: $-2=2(3)+b$. The point’s $-2$ isn’t $b$: its $x$-coordinate is $3$, not $0$.

Type $-2\sim 2(3)+b$ in Desmos. The $\sim$ tells Desmos to find the parameter $b$ that makes the equation true; under PARAMETERS, it shows $b=-8$. So the parallel line through $(3,-2)$ is $y=2x-8$. Choice D.