Tables: 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 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 much y changes for a one-unit increase in x.

The y-intercept (b) is the value of y when x is 0. It represents the starting point of the line on the y-axis.

To identify a linear equation from a table, check if the change in y is consistent for equal changes in x. If the differences are constant, the relationship is linear.

The equation is y = 2x + 3.

The slope is 2, calculated as (change in y) / (change in x) = 4 / 2.

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

To determine the equation, first calculate the slope (m) using the formula m = (y2 - y1) / (x2 - x1). Then use one of the points to solve for the y-intercept (b) and write the equation in the form y = mx + b.

The point (0, b) is the y-intercept of the line, where the line crosses the y-axis.

A proportional relationship is a specific type of linear relationship that passes through the origin (0,0), meaning that the ratio of y to x is constant.