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Linear equations are at the heart of algebra and are the first exposure students have to relationships between variables. These equations can be identified by their standard form, which is an equation of the first degree. This means that the highest power of the variable, often represented as 'x', is one.

The equation provided in the exercise, \(y = -2x\), is a perfect example of a linear equation. The equation implies that 'y' changes at a constant rate with respect to 'x'. For every single unit increase in 'x', 'y' decreases by 2 units. The beauty of linear equations lies in their simplicity and the fact that they graph as straight lines on the coordinate plane—a concept we will explore in more detail in the next section.

When it comes to graphing lines based on linear equations, understanding the significance of intercepts is fundamental. The x-intercept occurs where the graph crosses the x-axis, and the y-intercept is where the graph crosses the y-axis.

To graph a line like the equation from our exercise, \(y = -2x\), you typically need just two points. However, in this special case, both the x-intercept and y-intercept are at the same point, (0, 0). This means our line passes through the origin, which is also an intercept itself. It's interesting to note that the origin is the only point needed to graph this particular line. Knowing this, you draw a line through (0, 0) with a slope of -2, declining to the right along the graph. This is a simple representation of the concept of slope, which tells us the steepness and direction of a line.

The coordinate system, often referred to as the Cartesian coordinate system, is a two-dimensional plane divided by two perpendicular lines: the x-axis (horizontal) and the y-axis (vertical). Each point on this plane can be specified by an ordered pair of numbers, which represent its distances from the two axes.

What's great about the coordinate system is its ability to clearly show the relationship between two variables—what mathematicians call mapping. For instance, in our exercise, plotting the equation \(y = -2x\) involves marking the x-intercept and the y-intercept. However, both intercepts are (0, 0), placing the line's start at the very center of the coordinate system, known as the origin. The coordinate system allows us to understand the geometric representation of equations, making abstract algebraic concepts visibly tangible and easier to comprehend.