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The slope of a line is a crucial concept when working with linear equations. It essentially tells us how steep the line is. The slope, often represented by the letter \( m \), is defined as the ratio of the vertical change (rise) to the horizontal change (run) between two points on a line. This is usually expressed as the formula:

• \( m = \frac{\text{Rise}}{\text{Run}} \)

To calculate the slope from the equation \( 3x - 2y = 12 \), we first rearranged the equation into the slope-intercept form \( y = mx + c \). Solving gives us \( y = \frac{3}{2}x - 6 \). Here, \( m = \frac{3}{2} \), meaning for every 2 units you move to the right, you move up 3 units. This ratio helps us understand the direction and angle of the line in a coordinate system.

The y-intercept is the point where the line crosses the y-axis. This value is vital as it provides a starting point for drawing the graph of an equation. It is represented by the letter \( c \) in the slope-intercept form \( y = mx + c \).From our equation \( y = \frac{3}{2}x - 6 \), we identify the y-intercept as \( c = -6 \). This means that when the value of \( x \) is zero, the value of \( y \) is \(-6\). On a graph, this point is plotted at \((0, -6)\). Identifying this point was the first step in our graphing process, as it is where the line begins on the y-axis.

Linear equations represent straight lines when graphed on a coordinate plane. They are typically written in two forms - standard form and slope-intercept form.

• Standard form: \( Ax + By = C \)
• Slope-intercept form: \( y = mx + c \)

Transforming a linear equation from standard form to slope-intercept form is crucial for easily identifying the slope and y-intercept, aiding in graphing efforts. The original equation given \( 3x - 2y = 12 \) was in standard form. By solving for \( y \), we put it into slope-intercept form, \( y = \frac{3}{2}x - 6 \). This transformation helps us understand and visualize how lines behave on a graph.

Graphing lines can be straightforward once you have the slope-intercept form of an equation. This form provides all the necessary details to plot the line on a coordinate plane.To graph our line \( y = \frac{3}{2}x - 6 \):

• Start by plotting the y-intercept \((0, -6)\).
• Use the slope \( \frac{3}{2} \) to find a second point. From \((0, -6)\), move up 3 units and right 2 units, reaching another point.

These points direct us in drawing a straight line through them, representing the equation. Remember, the slope ensures the line's steepness is consistent, providing a visual representation of the linear equation.