Linear Relationships

A linear relationship is a relationship between two variables that can be represented by a straight line on a graph. It can be described by the equation of 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 steep the line is and the direction it goes (positive or negative).

The slope-intercept form of a linear equation is $$y = mx + b$$ , where $$m$$ is the slope and $$b$$ is the y-intercept.

The slope (m) is calculated using the formula $$m = \frac{y_2 - y_1}{x_2 - x_1}$$ .

The y-intercept is the point where the line crosses the y-axis. It is represented by the value of $$b$$ in the slope-intercept form $$y = mx + b$$ .

A line with a positive slope rises from left to right, indicating that as the x-value increases, the y-value also increases.

A line with a negative slope falls from left to right, indicating that as the x-value increases, the y-value decreases.

A slope of zero indicates a horizontal line, meaning there is no change in the y-value as the x-value changes.

An undefined slope indicates a vertical line, meaning the x-value does not change while the y-value can vary.

The slope can be identified by selecting two points on the line and using the formula $$m = \frac{y_2 - y_1}{x_2 - x_1}$$ .