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Understanding the slope and the y-intercept is crucial when graphing linear equations. These two key features will determine the unique straight line that represents the equation.

The slope, typically denoted as m, defines how steep the line is. It represents the rate at which the y value of a point changes per unit increase in the x value. For the equation given in the exercise, y = 3x - 5, the slope is 3. This means that for each step we move to the right along the x-axis, the y-value increases by 3 units.

Meanwhile, the y-intercept is where the line crosses the y-axis. It is written as 'c' in the equation y = mx + c and describes the starting point of the line when x is zero. In our equation, the y-intercept is -5, which is the point (0, -5) on the graph. This is the first point we plot on our graph and serves as the foundation upon which we build the line by using the slope.

Plotting points is a methodical process that provides a visual representation of the relationship between two variables described by an equation. Starting with the y-intercept, as we did with the point (0, -5), you then use the slope to find additional points.

You can think of plotting points as laying down breadcrumbs to mark your trail. Our slope is 3, which means for every horizontal movement of 1 unit, we make a vertical movement of 3 units up. From the y-intercept, we move right to (1, -5), then up to (1, -2), and plot this second point.

It's always good to plot more than two points to ensure accuracy when drawing your line. With each point plotted, you’re piece by piece, creating a path that the line will follow. Connect these points with a ruler for precision, and you've graphed your equation.

The slope of a line is a measure of its steepness and direction. Calculated as the 'rise over run,' it determines how much the line 'rises' vertically for each unit of 'run' horizontally.

In the context of our exercise, the slope of the line is 3. Visually, this slope creates a line that ascends rapidly because for every unit we move horizontally, the rise is three times as much. The slope would be different if, for instance, our equation were y = 2x - 5, our line would rise more slowly, 2 units up for each 1 unit across.

Finally, the sign of the slope also indicates the direction of the line. A positive slope, as in our exercise, means the line tilts upwards from left to right. Conversely, a negative slope would tilt downwards. By mastering the concept of the slope, students gain an intuitive understanding of how linear equations graphically represent various rates of change.