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The coordinate plane, sometimes referred to as the Cartesian plane, is a two-dimensional surface on which we can plot points, lines, and curves. It's made up of two perpendicularly intersecting lines called axes. The horizontal axis is known as the x-axis, and the vertical axis is called the y-axis. These axes divide the plane into four quadrants, which are used to define the location of points based on their distance from the origin — the point where both axes intersect.

When graphing linear equations, understanding the coordinate plane is crucial as it allows us to visualize relationships between variables. In our example, each pair of x and y values represents a point on this plane. By finding points for several values of x, we can draw a line that represents the equation on the plane.

Plotting points is a fundamental skill in graphing linear equations. The process involves identifying ordered pairs (x, y) and marking them on the coordinate plane. The first number of the pair, x, indicates how far to move horizontally from the origin. A negative x means moving left, while a positive x means moving right. The second number, y, tells us how far to move vertically, with negative values moving down and positive values moving up from the origin.

In our exercise, after substituting values of x into the equation to find y, we get pairs like (-3,-5) or (2,0). Each of these pairs is a point that can be plotted. By plotting all these points and connecting them with a straight line, we accurately represent the equation graphically.

A linear function is an algebraic equation that describes a straight line. The general form of a linear equation in two variables is given by: \( y = mx + b \), where \( m \) is the slope of the line, showing the steepness and direction of the line, and \( b \) is the y-intercept, the point where the line crosses the y-axis. In our exercise solution, the equation \( y = x - 2 \) is a linear function with a slope of 1 and a y-intercept of -2.

This means that as x increases or decreases by 1 unit, y increases or decreases by the same amount, which is why the points form a straight line. Understanding a linear function's slope and y-intercept is key when graphing, as it allows us to quickly plot two critical points and draw the line representing the function.

The substitution method is an effective way to solve for one variable in terms of another in an equation. This technique is particularly useful when graphing linear equations. When we graph an equation like \( y = x - 2 \), we're solving for y for different values of x. This process of substitution involves taking a known value of x, inserting it into the equation, and solving for the corresponding y value.

For instance, when we substitute x with -3, we get y equal to -5. This straightforward approach allows us to calculate several pairs of (x, y) that we then plot on the coordinate plane. These plotted points guide us in drawing the graph of the linear equation precisely.