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The slope is a fundamental concept in understanding the nature of a line on a graph. It indicates the steepness and direction of the line. A greater slope value means a steeper line, while a smaller slope indicates a flatter line. The slope is usually represented by the symbol 'm'.

To calculate the slope of a line from its equation in the form of \( y = mx + b \), we identify 'm'. For the line expressed as \( p(t) = 3t + 4 \), the slope is 3.

Perpendicular lines have a special relationship. Their slopes are negative reciprocals of each other. So, if one line has a slope of 3, the line perpendicular to it will have a slope of \( -\frac{1}{3} \). This is because the product of the slopes of two perpendicular lines is always -1.

The point-slope formula is a versatile tool used to find the equation of a line when you know a point on the line and the slope. The formula is expressed as \( y - y_1 = m(x - x_1) \). Here, \( m \) is the slope, and \( (x_1, y_1) \) are the coordinates of the given point.

In our exercise, we have the perpendicular slope \( m = -\frac{1}{3} \) and the point \( (3, 1) \). Plugging these into the point-slope formula gives us:

• \( y - 1 = -\frac{1}{3}(x - 3) \)

This equation represents a line passing through the point \( (3, 1) \) with a slope of \( -\frac{1}{3} \).

This step is crucial because it connects the slope and a particular point in a clear and direct way, helping to form the basis for deriving the final equation of the line.

The slope-intercept form of a line is \( y = mx + b \), where 'm' is the slope and 'b' is the y-intercept. This form is helpful for easily identifying the slope and where the line crosses the y-axis.

To convert the equation from point-slope to slope-intercept form, use algebraic manipulation. From our point-slope equation \( y - 1 = -\frac{1}{3}(x - 3) \), you need to:

• Distribute \( -\frac{1}{3} \) over \( (x - 3) \):
  \( y - 1 = -\frac{1}{3}x + 1 \)
• Add 1 to both sides to solve for \( y \):
  \( y = -\frac{1}{3}x + 2 \)

The new equation \( y = -\frac{1}{3}x + 2 \) offers a clearer view of the line's behavior. It shows the slope is \( -\frac{1}{3} \), aligning with the perpendicular nature to the original line, and it indicates the line crosses the y-axis at 2.