Algebra

Graph a line that is perpendicular to the given line. Determine the slope of the given line and the one you graphed in simplest form. The given line is a vertical line (x = -3) with an undefined slope. The perpendicular line would be a horizontal line with a slope of 0. Original Slope: undefined Perpendicular Slope: 0

Step-by-step solution with explanation

Final Answer

Original slope: undefined (vertical line x = −3). Perpendicular slope: 0 (any horizontal line y = c).

Step-by-step solution

1

Identify the given line and its slope

The given line is a vertical line passing through every point where x equals -3. Vertical lines have an undefined slope because the 'run' (horizontal change) is zero, and you cannot divide by zero.
2

Recall the perpendicular slope rule

Two lines are perpendicular when their slopes are negative reciprocals of each other. The special case: a vertical line (undefined slope) is always perpendicular to a horizontal line (slope = 0).
3

Write the perpendicular line equation

A horizontal line has the form y = c, where c is any number. No matter what x is, y stays the same — that gives a slope of zero. For example, y = 0 is the x-axis, which is perpendicular to x = -3.
4

Confirm slope of the perpendicular line

Pick any two points on a horizontal line, like (0, 0) and (5, 0). The rise is 0 and the run is 5, so slope = 0/5 = 0. This confirms the perpendicular line has a slope of 0.
5

State both slopes in simplest form

The given vertical line has an undefined slope. The graphed horizontal line has a slope of exactly 0. These are already in their simplest form — no fractions to reduce.

Understanding this problem

Learning Insight

Slope measures steepness as rise divided by run. A vertical line has zero run, making division impossible — that is why the slope is undefined, not infinite. A horizontal line has zero rise, making the slope exactly zero. These two directions are perfectly perpendicular (90° apart), which is why undefined slope and zero slope always go together as a perpendicular pair.

Quick Tip

Quick pattern to remember: vertical ↔ undefined slope, horizontal ↔ zero slope, and vertical lines are ALWAYS perpendicular to horizontal lines. If you see 'x = number,' the perpendicular line must be 'y = number.'

Common Mistake

Students often say a vertical line has a slope of zero instead of undefined, and confuse it with a horizontal line. Remember: zero slope means FLAT (horizontal), and undefined slope means STRAIGHT UP (vertical).