Question 233·Medium·Algebra

How are the slopes of two perpendicular lines related? If one slope is $2$, what is the negative reciprocal?

Once you know the slope of line $r$ and that it passes through $(4,-2)$, use point-slope form $y - y_1 = m(x - x_1)$ and then solve for $y$.

Type y = 2x + 1 into Desmos to graph the given line and see its slope visually.

Add the point A = (4, -2) in Desmos so you can easily see which lines pass through this point.

Enter each option as a separate line in Desmos: y = -1/2x + 4, y = -2x - 1, y = -1/2x, and y = 2x - 4. Observe which lines go through point A and how they are oriented relative to the original line.

Look for the line that both passes through point A and meets the original line at a right angle (perpendicular). Use the slopes shown by the graphs or the steepness and direction of the lines to confirm your choice.

The equation of the given line is $y = 2x + 1$, which is in slope-intercept form $y = mx + b$.

• The slope $m$ of this line is $2$.

Slopes of perpendicular lines are negative reciprocals of each other.

• The reciprocal of $2$ is $\frac{1}{2}$.
• The negative reciprocal of $2$ is therefore $-\frac{1}{2}$. So line $r$ must have slope $-\frac{1}{2}$.

Use the point-slope form with point $(4,-2)$ and slope $-\frac{1}{2}$:

Simplify:

Subtract 2 from both sides:

This matches answer choice C) $y = -\frac{1}{2}x$, so that is the equation of line $r$.