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A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable. Linear equations can be written in many forms, including slope-intercept form, which is particularly useful for graphing. This form is expressed as:

\[ y = mx + b \]

where \( m \) is the slope of the line and \( b \) is the y-intercept, which is where the line crosses the y-axis. Solving the exercise involved converting the given equation to this form, which makes it straightforward to identify these key characteristics and visualize them on a graph.

Graphing a linear equation involves plotting points on a coordinate plane and connecting them with a straight line. The slope-intercept form facilitates graphing since you can easily plot the y-intercept (\( b \)) on the y-axis. This point is where the line crosses the y-axis. Next, use the slope (\( m \)) which is the 'rise over run' (the amount the line goes up divided by the amount it goes right) to find another point on the line.

Using the slope \( 2 \) from our exercise as an example, we 'rise' up 2 units and 'run' to the right 1 unit from the y-intercept to find a second point. Draw a straight line through these points, and you've successfully graphed the linear equation.

The slope of a linear equation represents the steepness or incline of the line and is usually denoted by \( m \). It shows the change in the y-value for each unit change in the x-value along the line. A positive slope means the line rises as it moves from left to right, while a negative slope indicates it falls.

The y-intercept is the point where the line crosses the y-axis, represented by \( b \) in the slope-intercept form. It's crucial when graphing because it provides a starting point on the graph where the line will pass through.

In our exercise, the slope-intercept form \( y = 2x - 7 \) immediately tells us the slope is 2, indicating that for each step right on the x-axis, the line rises by 2 steps. The y-intercept is -7, placing our starting point below the origin at -7 on the y-axis.