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What Is Slope‑Intercept Form?

The slope‑intercept form is the most intuitive representation of a linear equation. It is written as:

y = mx + b

where m is the slope and b is the y‑intercept. This format instantly reveals two key characteristics of the line: how steep it is and where it crosses the y‑axis.

Key Definitions

• Slope (m): Measures the line’s steepness and direction. Positive values mean the line rises left to right; negative values mean it falls.
• Y‑Intercept (b): The point where the line meets the y‑axis (x = 0). For example, in y = 2x + 5 the intercept is (0, 5).

Graphing a Line

To plot a line in slope‑intercept form, you need two points:

1. Plot the y‑intercept (0, b).
2. Find the x‑intercept by setting y = 0 and solving for x. For y = 2x + 5: 0 = 2x + 5 → x = -5/2, giving the point (-2.5, 0).
3. Draw the line through these points.

Other Common Forms

• Standard Form: Ax + By = C. Example: 10x + 2y = 1.
• Point‑Slope Form: y – y1 = m(x – x1). Example: y – 2 = 5(x – 7).

Creating Parallel Lines

Parallel lines share the same slope but have different y‑intercepts. Keep the slope from the original line and change b. For y = 3.5x + 20, a parallel line could be y = 3.5x + 14.

Creating Perpendicular Lines

Perpendicular lines have slopes that are negative reciprocals. If the original slope is m, the perpendicular slope is –1/m. For y = 3.5x + 20, the perpendicular slope is –2/7, so any line y = (-2/7)x + b will be perpendicular.

Finding a Line Through a Specific Point

Given a point (x₁, y₁) and a slope m, substitute into the slope‑intercept form to solve for b. Example: To find a line with slope 3.5 passing through (1, 1): 1 = 3.5(1) + b → b = –2.5, giving y = 3.5x – 2.5.