Modeling Linear Relationships and Slopes

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

The slope (m) 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 (m) is calculated using the formula: m = (y2 - y1) / (x2 - x1).

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

A positive slope indicates that as the independent variable (x) increases, the dependent variable (y) also increases.

A negative slope indicates that as the independent variable (x) increases, the dependent variable (y) decreases.

The standard form of a linear equation is Ax + By = C, where A, B, and C are integers, and A should be non-negative.

A linear equation can be identified from a table of values if the differences between consecutive y-values divided by the differences between consecutive x-values are constant.

A function is a relation where each input (x-value) has exactly one output (y-value). A non-function has at least one input that corresponds to multiple outputs.

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