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Graphing linear equations involves plotting points on a graph where the x and y values form a straight line. The equation of a line is commonly presented in the slope-intercept form, which is \[ y = mx + b \]where \( m \) represents the slope and \( b \) is the y-intercept—the point where the line crosses the y-axis.

To graph the equation \( y = 5x - 10 \), we start by finding points that satisfy this equation. These points are obtained by selecting different x-values and computing the corresponding y-values. Once the points are plotted on a graph, they collectively depict a straight line.

Here is the simple process to create the graph:

• Select some x-values you want to explore.
• Substitute these values into the equation to calculate y-values.
• Plot the (x, y) pairs on the coordinate plane.
• Draw a line through the points, ensuring it extends across the graph.

This visual representation allows us to easily understand the relationship between x and y in a linear equation.

The Cartesian coordinate system is a mathematical framework used to plot points and graph equations. It comprises two perpendicular lines: the x-axis (horizontal) and the y-axis (vertical), which intersect at the origin (0,0). Points are defined by ordered pairs (x, y) that specify their location on this plane.

When plotting on a Cartesian plane, remember:

• X-axis: The horizontal axis where values increase to the right and decrease to the left of the origin.
• Y-axis: The vertical axis where values increase upwards and decrease downwards from the origin.
• Quadrants: The plane is divided into four quadrants, each identified by the positive or negative signs of the x and y coordinates.

In our solution, when plotting points or reading a graph, understanding the Cartesian coordinate system helps determine where each point, like (2, 0) or (-1, -15), should be positioned.

Function tables are a practical way to organize information about a function. They list input values and their corresponding output values, aiding in identifying patterns and relationships in functions like the linear equation given.

To construct a function table for the linear equation \( y = 5x - 10 \), perform the following steps:

• Choose a set of x-values. In this exercise, we used \{-2, -1, 0, 1, 2\}.
• Substitute each value into the equation to find the respective y-values.
• Organize these (x, y) pairs into a table format.

This method simplifies the task of plotting graphs, as it provides a clear, organized overview of x and y values to be plotted. The role of function tables is to streamline calculations and act as a handy reference when graphing or solving linear equations.