The decision and its conditions
Parallel nonvertical lines share a slope, while perpendicular nonvertical lines have slopes whose product is negative one. A point supplies the remaining condition needed to determine an intercept. Coordinate distances combine horizontal and vertical changes with the Pythagorean theorem; signs disappear only after correctly taking differences and squaring. An x-intercept occurs when y is zero, unlike a y-intercept where x is zero. Use the geometric condition to choose the algebraic operation instead of treating every number in an equation as an interchangeable coordinate.
Worked example
A nonvertical line has slope 2/3. What is the slope of a line perpendicular to it? Perpendicular nonvertical slopes multiply to negative one. Take the negative reciprocal of 2/3. The result is -3/2, and 2/3×(-3/2)=−1.
A contrasting decision
A line parallel to y=3x−7 passes through (3,13). In y=3x+b, find b. Parallel lines share slope 3. Substitute the new point: 13=3×3+b. Thus b=13−9=4.
Diagnose a competing method
The first example’s tempting answer 3/2 follows this mistake: Take a reciprocal but omit the negative sign. Parallel lines preserve slope; perpendicular lines use a negative reciprocal.
Transfer before revealing
Find the straight-line distance between points (-2,3) and (7,15). Pause and identify the requested quantity. Solution: The horizontal change is 9, and the vertical change is 12. These perpendicular changes form a right triangle; square, add and take a root. The distance is 15 coordinate units.
Check the boundary of the method
An x-intercept occurs when y is zero, unlike a y-intercept where x is zero. Keep the slope from the given line, but recover a new intercept from the new point. Compare the answer with the question’s units and conditions. One correct example does not prove an identity or a statistical generalization.
A nonvertical line has slope 2/3. What is the slope of a line perpendicular to it?
- Perpendicular nonvertical slopes multiply to negative one.
- Take the negative reciprocal of 2/3.
- The result is -3/2, and 2/3×(-3/2)=−1.
Compare the other answer choices
Add the numerator and denominator instead of using the perpendicular-slope rule.
Take a reciprocal but omit the negative sign.
Negate the original slope then divide by its denominator again instead of inverting it.
Explain your thinking.
Optional written extension. Self-assessed only; it does not change your automatic practice result. These are teaching criteria, not official awarded marks.
Show the model and intermediate calculations, then refute one competing method. This written extension is not automatically scored.
This draft stays on this page only. Use paper if you want to keep your working.
Try it, then explain it.
Complete a short set without looking at the example. After each answer, explain which operation preserved the relationship and why the result makes sense.
Sources & authorship
Original Slope explanations and examples. References support the concepts; they are not the source of copied questions. Independent educator review is pending.
GED Mathematical Reasoning Assessment Guide for Educators