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The slope of a line is a measure of its steepness and direction. In the slope-intercept form of a linear equation, represented by \( y = mx + b \), \( m \) is the slope. This value indicates how much the line rises or falls as we move along the x-axis. For the given equation, after converting it into slope-intercept form, we find that the slope \( m \) is \( \frac{6}{7} \). This tells us that for every 7 units we move horizontally—meaning along the x-axis—the line goes up by 6 units vertically. Slope has a few key characteristics:

• Positive slope: The line rises from left to right.
• Negative slope: The line falls from left to right.
• Zero slope: The line is horizontal.
• Undefined slope: The line is vertical.

In our case, a slope of \( \frac{6}{7} \) indicates a mild upward trajectory as we move from left to right.

The y-intercept of a line is the point where the line crosses the y-axis. In the slope-intercept form \( y = mx + b \), \( b \) represents the y-intercept. It is the value of \( y \) when \( x = 0 \). This is because, at the y-intercept, the line meets the y-axis, where all points have an x-coordinate of 0.For the equation \( y = \frac{6}{7}x - 2 \), the y-intercept \( b \) is -2. This means that the line crosses the y-axis at the point (0, -2).

If you were to graph this, you'd start by plotting the point on the y-axis directly 2 units below the origin. This is an essential step in the process of graphing, providing a reliable starting point for plotting the rest of the line.

Graphing linear equations involves translating an equation in slope-intercept form, \( y = mx + b \), onto the coordinate plane. This process visually represents the line described by the equation.

To graph the line from the equation \( y = \frac{6}{7}x - 2 \), begin by plotting the y-intercept at (0, -2). This gives the initial point on the graph.

Next, use the slope, which is \( \frac{6}{7} \), to determine the direction and steepness of the line. From the y-intercept, move 6 units up ("rise") and 7 units to the right ("run"), putting you at the point (7, 4). Plotting this second point enables you to draw a straight line through both points, extending in both directions.

Remembering these key steps:

• Identify and plot the y-intercept \( (0, b) \).
• Use the slope for direction and spacing \( (rise, run) \).
• Draw a line through the points.

This method ensures a precise representation of the linear equation on the graph.