91Ó°ÊÓ

The slope-intercept form of a linear equation is one of the simplest and most popular ways to express a line. It is given by the equation \(y = mx + b\), where:

• \(m\) is the slope of the line.
• \(b\) is the y-intercept, which is the point where the line crosses the y-axis.

This form is immensely useful because it provides direct information about the slope and y-intercept.
Understanding the slope, \(m\), is crucial—the slope tells us the steepness of the line and whether it is ascending or descending as we move from left to right. For a slope of \(-2\), as given in the exercise, the line falls 2 units for every 1 unit it moves to the right.
The y-intercept, \(b\), helps establish the line's position. If a line passes through the origin, then \(b = 0\). For our specific equation, substituting the slope correctly yields:
Given: \(-2(x - 2)\), solving gives: \(y = -2x + 4\), matching the slope-intercept form's expectations.

The x-intercept of a line is the point where the line crosses the x-axis. At this point, the value of \(y\) is zero. To find the x-intercept from a linear equation, we set \(y = 0\) and solve for \(x\).
In our exercise, the x-intercept is given as \(x=2\). This means when \(x=2\), the line crosses the x-axis and thus, \(y = 0\).
To verify or find an x-intercept, substitute \(0\) for \(y\) in the linear equation. Doing this provides an opportunity to double-check the results. Using our equation: \(y = -2x + 4\), if \(y = 0\), substituting gives:

• \(0 = -2(2) + 4\)
• \(0 = -4 + 4\)
• The expression is balanced, confirming the point \((2, 0)\).

Understanding intercepts like \(x = 2\) is foundational in plotting and interpreting lines in coordinate geometry.

Another method to express the equation of a line is the point-slope form. It is especially practical when you know the slope of the line and a specific point through which the line passes. This form is given as:\(y - y_1 = m(x - x_1)\), where:

• \(m\) is the slope.
• \((x_1, y_1)\) is a known point on the line.

In our exercise, the slope \(m = -2\) and the point \((2, 0)\) are given. This information fits perfectly into the point-slope format:\(y - 0 = -2(x - 2)\).
Breaking it down, \(x_1 = 2\) and \(y_1 = 0\) are plugged into the formula. This format is valuable because it helps quickly transition to the slope-intercept form by just simplifying:

• Solving \(y = -2(x - 2)\) leads to: \(y = -2x + 4\), seamlessly converting to the slope-intercept form \(y = mx + b\).

The point-slope form directly ties the characteristics of a line to specific, tangible points in coordinate geometry.