91Ó°ÊÓ

The slope-intercept form of a linear equation is represented as \(y = mx + b\). In this form, \(m\) represents the slope of the line, while \(b\) represents the \(y\)-intercept, the point where the line crosses the \(y\)-axis. This form is particularly beneficial when you want to quickly identify the slope and \(y\)-intercept without additional calculations.

Understanding slope-intercept form helps in graphing linear equations easily. You can plot the \(y\)-intercept on a graph and then use the slope to determine the direction and steepness of the line from that point.

For example, in the equation \(f(x) = -0.5x + 1\), the slope \(-0.5\) indicates that the line will decrease as \(x\) increases, and it will intersect the \(y\)-axis at \(b = 1\).

Perpendicular lines have a unique relationship based on their slopes. If two lines are perpendicular, the product of their slopes equals \(-1\). This is a crucial fact to remember when dealing with perpendicular lines.

For instance, in the given problem, the line perpendicular to our target line has a slope of \(2\). To find the slope of the target line that is perpendicular to this, we utilize the relationship formula:

• \(m_1 \cdot m_2 = -1\)
• where \(m_1 = 2\) (slope of the given line) and \(m_2 = -0.5\) (slope of the perpendicular line)

This relationship ensures the lines will intersect at a \(90^\circ\) angle. Understanding the perpendicular slope relationship is vital for constructing the equation of a line that meets specific geometric criteria.

The slope of a line, \(m\), measures its steepness and direction. It can be calculated using the formula: \(m = (y_2 - y_1)/(x_2 - x_1)\), where \((x_1, y_1)\) and \((x_2, y_2)\) are two distinct points on the line.

In the exercise, we calculated the slope of the line with intercepts \((2,0)\) and \((0,-4)\). Using the slope formula, the calculation was:

• \(m = (-4 - 0) / (0 - 2)\)
• \(m = 2\)

Determining the slope is essential for writing equations of lines and understanding their graphical representations. A positive slope means the line rises as it moves right, while a negative slope signifies a decline.