Straight Lines

Every straight line on the Cartesian plane can be described by an equation. Understanding these equations lets you predict where a line goes, whether two lines will meet, and at what angle.

Slope (Gradient)

The slope measures how steep a line is -- how much it rises or falls as you move right.

Slope Formula: m = (y2 - y1) / (x2 - x1)

• Positive slope: line goes uphill (left to right)
• Negative slope: line goes downhill
• Zero slope: horizontal line
• Undefined slope: vertical line

Example: Slope through (1, 3) and (4, 9): m = (9 - 3)/(4 - 1) = 6/3 = 2

Forms of Line Equations

Slope-Intercept Form

y = mx + c where m is the slope and c is the y-intercept.

Example: y = 2x + 3 has slope 2 and crosses the y-axis at (0, 3).

Point-Slope Form

When you know the slope m and a point (x1, y1) on the line:

y - y1 = m(x - x1)

Worked Example: Find the equation of a line with slope 3 passing through (2, 5).

y - 5 = 3(x - 2), which simplifies to y = 3x - 1

Parallel and Perpendicular Lines

Parallel lines have the same slope: m1 = m2

Perpendicular lines have slopes that are negative reciprocals: m1 x m2 = -1

Example: A line has slope 2. A parallel line also has slope 2. A perpendicular line has slope -1/2.

Worked Example: Find the equation of a line perpendicular to y = 3x + 1 passing through (6, 2).

Perpendicular slope = -1/3. Using point-slope form: y - 2 = -1/3(x - 6), so y = -x/3 + 4

Nepal Connection: Road engineers use slope calculations when designing highways through Nepal's hilly terrain. A road with too steep a slope is dangerous; too gentle adds unnecessary length and cost.

Key Takeaways

• Slope = rise/run = (y2 - y1)/(x2 - x1)
• Slope-intercept form (y = mx + c) is great for graphing
• Point-slope form is best when you know a point and the slope
• Parallel lines: equal slopes. Perpendicular lines: slopes multiply to -1