slopeintercept

Slope-intercept form is a common way to express the equation of a straight line in a two-dimensional Cartesian coordinate system. It is written as y = mx + b, where m is the slope and b is the y-intercept.

The slope m describes the line's steepness, equal to the ratio of the change in y to the change in x between any two points on the line. The y-intercept b is the point where the line crosses the y-axis, at x = 0, giving the coordinate (0, b). If b is positive, the line crosses above the origin; if negative, below.

To graph from slope-intercept form, plot the point (0, b) on the y-axis. Then use the slope m to determine a second point: move horizontally by 1 unit and vertically by m units (up if m is positive, down if negative). Draw the line through the two points.

Vertical lines cannot be represented in slope-intercept form because their slope is undefined. In that case, other forms such as the standard form are used.

Converting other forms to slope-intercept form is common. From point-slope form y - y1 = m(x - x1), expanding gives y = mx + (y1 - m x1). From standard form Ax + By = C (with B ≠ 0), solving for y yields y = (-A/B)x + C/B.

Example: y = 3x + 2 has slope m = 3 and y-intercept b = 2. It crosses the y-axis at (0, 2); points include (1, 5) and (2, 8).

Applications include modeling linear relationships, quick graphing, and predicting y for given x values in various real-world contexts.