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Creating a table of values is a fundamental step in graphing linear equations. Essentially, it's a convenient way to organize pairs of numbers that are solutions to the equation. The process begins by selecting various x-values, which can be any numbers you choose, though typically integers are used for simplicity. Then, for each x-value, the corresponding y-value is calculated using the equation of the line.

For the given equation, \(3x - 2y = 6\), one might choose x-values like -1, 0, and 1. Plugging these into the equation yields the y-values after some arithmetic. The resulting pairs (x,y) are points that lie on the graph of the equation. For instance, if x is -1, calculation gives y as 1.5, creating the point (-1,1.5).

Using a variety of x-values is wise because it helps confirm the line's consistency and whether the graphed points align correctly. It's like plotting snapshots of the equation's behavior across the Cartesian plane.

The slope-intercept form is a straightforward way to express a linear equation and is written as \( y = mx + b \). In this formula, \( m \) represents the slope of the line, and \( b \) signifies the y-intercept, the point where the line crosses the y-axis.

The convenience of this form lies in its direct depiction of the graph's characteristics: \( m \) tells us how steep the line is and the direction it tilts, while \( b \) provides a starting point for plotting. Convert the given equation \(3x - 2y = 6\) to slope-intercept form by isolating \( y \) yields \( y = \frac{3}{2}x - 3 \), making the slope \( \frac{3}{2} \) and the y-intercept -3. The slope suggests that for each unit increase in x, y increases by 1.5 units, and the graph crosses the y-axis at (0, -3).

After determining points through the table of values, plotting them on the Cartesian plane is the next critical step. The Cartesian plane consists of a horizontal x-axis and a vertical y-axis where each point is represented by an ordered pair (x,y).

To plot a point, start at the origin (0,0), move horizontally to the x-value, and then vertically to the y-value. For example, the point (0,-3) lies directly below the origin on the y-axis at three units distance because its x-value is 0. Plotting the points from our table - (-1,1.5), (0,-3), (1,-4.5) - and then drawing a line through them establishes the graph of the equation.

The lined-up points should form a straight line on the plane, revealing the linear relationship between x and y as dictated by our original equation.

Isolating variables is a technique used to simplify equations and solve for one variable in terms of others. For graphing, focusing on y is typical because it allows us to use the slope-intercept form.

To isolate y in the equation \(3x - 2y = 6\), we perform operations that reverse the equation's current effects on y. First, subtract \(3x\) from both sides to move it away from y, yielding \( -2y = -3x + 6\). Next, divide everything by -2 to 'free' y, leading us to \( y = \frac{3}{2}x - 3\). Now, y is expressed only in terms of x, and its graphing via table of values or slope-intercept becomes substantially easier.

This manipulation highlights an essential algebraic skill: rearranging equations to facilitate the visualization of solutions on a graph.