A linear relationship is a type of relationship between two variables in which the change in one variable is proportional to the change in the other variable. The equation of a linear relationship is typically written in the form y = mx + b, where y represents the dependent variable, x represents the independent variable, m represents the slope of the line, and b represents the y-intercept.

**Slope:**The slope of a line represents the rate of change of the dependent variable with respect to the independent variable. It is calculated as the ratio of the vertical change (change in y) to the horizontal change (change in x) between two points on the line.**Y-Intercept:**The y-intercept is the point where the line intersects the y-axis. It represents the value of the dependent variable when the independent variable is zero.**Graphing Linear Equations:**To graph a linear equation, plot the y-intercept on the y-axis and then use the slope to find additional points to create the line.**Equation of a Line:**The general form of the equation of a line is y = mx + b, where m is the slope and b is the y-intercept.**Using Linear Equations:**Linear equations can be used to model and solve real-world problems involving two variables that have a constant rate of change.

- Understand the concept of slope and how it relates to the steepness of a line.
- Practice identifying and interpreting the y-intercept of a linear equation.
- Work on graphing linear equations and understanding how the slope and y-intercept affect the shape of the line.
- Practice solving word problems using linear equations to model real-world situations.
- Review and understand the different forms of linear equations, such as slope-intercept form and standard form.

Given the linear equation y = 2x + 3, find the slope and y-intercept, then graph the line.

**Solution:**

Slope (m) = 2

Y-intercept (b) = 3

To graph the line, plot the point (0, 3) on the y-axis and use the slope to find additional points to create the line.

You can also solve real-world problems using linear relationships. Here's an example:

**Problem:**

A car rental company charges a flat fee of $30 plus $0.20 per mile. Write a linear equation to represent the total cost (C) in terms of the number of miles driven (m).

**Solution:**

The linear equation is C = 0.20m + 30, where 0.20 is the cost per mile and 30 is the flat fee.

Study GuideLinear relationships Worksheet/Answer key

Linear relationships Worksheet/Answer key

Linear relationships Worksheet/Answer key

Linear relationships

Data Analysis and Probability (NCTM)

Develop and evaluate inferences and predictions that are based on data.

Make conjectures about possible relationships between two characteristics of a sample on the basis of scatterplots of the data and approximate lines of fit.

Grade 8 Curriculum Focal Points (NCTM)

Algebra: Analyzing and representing linear functions and solving linear equations and systems of linear equations

Students use linear functions, linear equations, and systems of linear equations to represent, analyze, and solve a variety of problems. They recognize a proportion (y/x = k, or y = kx) as a special case of a linear equation of the form y = mx + b, understanding that the constant of proportionality (k) is the slope and the resulting graph is a line through the origin. Students understand that the slope (m) of a line is a constant rate of change, so if the input, or x-coordinate, changes by a specific amount, a, the output, or y-coordinate, changes by the amount ma. Students translate among verbal, tabular, graphical, and algebraic representations of functions (recognizing that tabular and graphical representations are usually only partial representations), and they describe how such aspects of a function as slope and y-intercept appear in different representations. Students solve systems of two linear equations in two variables and relate the systems to pairs of lines that intersect, are parallel, or are the same line, in the plane. Students use linear equations, systems of linear equations, linear functions, and their understanding of the slope of a line to analyze situations and solve problems.

Connections to the Grade 8 Focal Points (NCTM)

Data Analysis: Building on their work in previous grades to organize and display data to pose and answer questions, students now see numerical data as an aggregate, which they can often summarize with one or several numbers. In addition to the median, students determine the 25th and 75th percentiles (1st and 3rd quartiles) to obtain information about the spread of data. They may use box-and-whisker plots to convey this information. Students make scatterplots to display bivariate data, and they informally estimate lines of best fit to make and test conjectures.