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The Cartesian coordinate system is a framework used to plot mathematical functions and equations, including inequalities, on a two-dimensional plane. This system uses two axes: the horizontal axis (x-axis) and the vertical axis (y-axis). Each point in this system is represented by an ordered pair

• x-coordinate: indicating horizontal position
• y-coordinate: indicating vertical position

Setting up a Cartesian coordinate system involves drawing two perpendicular lines, labeling them, and marking intervals with numbers along each axis. The point where these axes intersect is the origin, which is denoted as (0,0). By plotting equations and inequalities in this system, we can visually interpret solutions and relationships between variables. This helps in recognizing the regions that satisfy certain conditions, like inequalities.

Linear equations are fundamental in algebra and involve variables raised to the power of one, resulting in a graph that forms a straight line. A typical form of a linear equation is\[ y = mx + b \]where:

• m is the slope of the line
• b is the y-intercept, where the line crosses the y-axis

In the case of the exercise at hand, the linear equation is\[ y = \frac{1}{3}x \]This has no constant term (b = 0), meaning the line passes through the origin. Understanding how to represent a linear equation on a Cartesian plane allows us to visualize the relationship between x and y values and subsequently determine solutions to equations and inequalities by shading appropriate regions.

The slope-intercept form is a simple way to express linear equations, prominently displaying important information about the line.In this form:\[ y = mx + b \]- Slope (m): The ratio of the change in y to the change in x, also known as "rise over run." It indicates how steep the line is.- Y-intercept (b): It shows the point where the line crosses the y-axis. In our exercise, since b is 0, the line crosses the origin.Using the slope-intercept form helps quickly graph the line by:1. Starting at the y-intercept on the graph.2. Using the slope to determine the direction and angle of the line (rise over run).This form simplifies graphing because it directly shows how different x values affect y, making it easier to draw and extend the line across the graph. Familiarity with the slope-intercept form is crucial for analyzing and solving linear inequalities and equations.