Parallel and perpendicular lines

Why this exists

A drawing shows only a few squares of each line, and lines go on for ever. The lines y = 3x + 7 and y = 3.01x + 2 look parallel on any ordinary sheet of graph paper, yet they cross at (500, 1507), far off the page. Looking cannot decide whether two lines meet.

The equations can. A line that is not vertical has a Slope, the amount it rises for each step to the right, and a y-intercept, the height where it crosses the y-axis. Same slope and different intercepts: the lines never meet. Different slopes: they cross, exactly once. The same comparison tells how many solutions a system of two equations has.

Definition

Two different lines in a plane are parallel when they never meet, however far they are extended. In symbols, \ell_1 \parallel \ell_2 says that line \ell_1 is parallel to line \ell_2. This is the parallel lines of geometry, now read through coordinates.

For two lines that are not vertical, written in slope-intercept form

the lines are parallel exactly when

If the slopes match and the intercepts match too, the two equations describe one and the same line. Some books count a line as parallel to itself; this lesson does not, so that “parallel” matches “no solution” exactly.

A vertical line has no slope, so vertical lines need their own statement: x = h and x = k are parallel exactly when h \ne k.

Intuition

Stand on one line at some value of x and measure straight up or down to the other line. Call that length the vertical gap at x. Two lines meet exactly where the vertical gap is 0, because there they have the same y for the same x.

Figure 1 draws \displaystyle y = \frac{1}{3}x + 2 and \displaystyle y = \frac{1}{3}x - 1. At x = 0 the gap is 2 - (-1) = 3. Take one step to the right. Each line rises by its slope, and both slopes are \displaystyle \frac{1}{3}, so the gap is still 3. The same happens at every step, to the right or to the left, so the gap is 3 at every x, never 0, and the lines never meet.

Figure 2 keeps \displaystyle y = \frac{1}{3}x + 2 but swaps the lower line for the steeper y = x - 3. Now each step to the right lifts the steeper line by 1 and the other by only \displaystyle \frac{1}{3}, so the gap closes by \displaystyle 1 - \frac{1}{3} = \frac{2}{3} per step:

At x = 7.5 the gap reaches 0 and the lines cross at (7.5, 4.5). To the right of that point the steeper line is on top and keeps pulling away; to the left the gap only grows. So the lines cross once, and only once.

Both pictures follow one rule. Each step to the right changes the gap by the difference of the slopes, m_1 - m_2. When the slopes are equal that difference is 0, so the gap is frozen at the difference of the intercepts. When the slopes differ, the gap changes at a steady rate that is not 0, and such a quantity passes through 0 exactly once.

Perpendicular lines

The other special way for two lines to sit is at a right angle, and slope triangles show what that does to their slopes. In Figure 3 the line through V has slope \displaystyle \frac{1}{3}: its slope triangle runs 3 to the right and rises 1, ending at P.

Give the whole picture a quarter turn counterclockwise about V. The turned line is at a right angle to the first, and the triangle turns with it. The run of 3, which pointed right, now points up: it has become a rise of 3. The rise of 1, which pointed up, now points left: it has become a run of -1. So the turned line has slope

Rise and run swapped places, which flips the fraction, and one of them changed sign. In the same way any slope \displaystyle \frac{a}{b} turns into \displaystyle -\frac{b}{a}, its negative reciprocal, and the two slopes multiply to -1; here \displaystyle \frac{1}{3} \cdot (-3) = -1. Lines at a right angle are called perpendicular, written \ell_1 \perp \ell_2. The lesson on slopes of perpendicular lines proves that every perpendicular pair behaves like this, and handles the horizontal and vertical cases this short section leaves aside.

Results

Worked examples

Boundaries

Vertical lines have no slope. The test “same slope, different intercept” needs both lines in the form y = mx + b. Two vertical lines are compared through x = h and x = k instead, and a vertical line is never parallel to a line that has a slope.

Slopes must match exactly. Rounding can hide a difference: 0.33 is not \displaystyle \frac{1}{3}, and lines with those slopes whose intercepts differ by 1 meet 300 units to one side of the y-axis, since the gap changes by only \displaystyle \frac{1}{3} - 0.33 = \frac{1}{300} per step. Keep slopes as exact fractions.

The vertical gap is not the distance between the lines. It is measured straight up, not straight across. The shortest distance between two slanting parallel lines is measured along a perpendicular and is smaller; that is the distance from a point to a line. Only for horizontal lines are the two the same.

The test belongs to the plane. In space two lines can miss each other without being parallel, one passing above the other, and a line in space has no single slope.

Where this goes next

Solving systems by graphing is the nearest use: slopes and intercepts say whether two line equations have one common solution, none, or infinitely many before any solving starts. Writing a parallel or perpendicular line through a point turns the third example into a general move.

In geometry, equal slopes prove sides parallel from coordinates, a step in classifying a figure from its corners, and parallel lines cut by a transversal add the angle facts. In calculus the tangent and normal lines to a curve use a slope and its negative reciprocal.

Exercises

Parallel, crossing, or the same line

Decide whether the lines in each pair are parallel, cross, or are one line. If they cross and the point is easy to read off, give it.

1. y = 2x + 9 and y = 2x - 1
2. \displaystyle y = -\frac{3}{5}x + 4 and \displaystyle y = -\frac{5}{3}x + 4
3. 3x - y = 7 and y = 3x + 2
4. 2x + 8y = 16 and x + 4y = 8
5. y = -2 and y = 5
6. x = 3 and y = 3
7. The line through (0, 1) and (4, 7), and the line through (-2, -5) and (2, 1)

Writing a parallel line

Write an equation of each line, in slope-intercept form unless the line is vertical.

8. Through (-6, 1), parallel to \displaystyle y = \frac{1}{3}x - 5.
9. Through (5, -2), parallel to 4x + 5y = 1.
10. Through (-1, 7), parallel to (a) y = 3, (b) x = 3.

Perpendicular slopes

11. Are the lines \displaystyle y = \frac{3}{8}x - 1 and 8x + 3y = 6 perpendicular?

Using the idea

12. On a map grid, one rail of a straight railway track lies along 2x - 5y = -15. The other rail is parallel to it and passes through (5, 4). Find its equation and the vertical gap between the rails.
13. For which value of k are the lines y = (k - 1)x + 4 and y = 3kx - 2 parallel?
14. Without solving, decide how many solutions the system y = 6x - 1, 12x - 2y = 5 has.

Reasoning

15. Elowen draws y = 3x - 2 and y = 2.99x + 1 for x from 0 to 10, and the lines look parallel. Are they? If they meet, find where, and explain why the drawing could not show it.
16. Lines p and q are both parallel to a line r, and none of the three is vertical. Explain why p and q are parallel to each other or are the same line.