Linear Relationships

A linear relationship is a relationship between two variables that can be represented by a straight line on a graph, typically expressed in the form y = mx + b, where m is the slope and b is the y-intercept.

The slope of a line represents the rate of change of the dependent variable (y) with respect to the independent variable (x). It indicates how much y changes for a one-unit increase in x.

A relationship is proportional if it can be represented by a straight line that passes through the origin (0,0). In a proportional relationship, the ratio of y to x is constant.

The y-intercept is the value of y when x is 0. It is the point where the line crosses the y-axis.

You can use the point-slope form of the equation: y - y1 = m(x - x1), where (x1, y1) is the point on the line and m is the slope.

A proportional relationship passes through the origin and has a constant ratio between y and x, while a non-proportional relationship does not pass through the origin and may have a different ratio.

The slope can be identified by selecting two points on the line and calculating the rise (change in y) over the run (change in x) between those points.

This equation represents the slope-intercept form of a linear equation, where m is the slope and b is the y-intercept.

To convert from standard form (Ax + By = C) to slope-intercept form (y = mx + b), solve for y in terms of x.

A positive slope indicates that as x increases, y also increases, while a negative slope indicates that as x increases, y decreases.