91Ó°ÊÓ

One of the easiest ways to express the equation of a straight line is through the slope-intercept form, which is represented as \( y = mx + b \). In this formula, \( m \) stands for the slope of the line and represents how steep the line is. The variable \( b \) represents the y-intercept, the point where the line crosses the y-axis. For example, if a line is described by the equation \( y = 3x - 2 \), its slope is 3, and it crosses the y-axis at (0, -2).
Understanding the slope-intercept form is critical for quickly sketching graphs of lines and for solving various algebra problems involving linear equations. It is a straightforward approach that directly gives two characteristic properties of a line: its steepness and the initial value where it intersects the y-axis.

The x-intercept of a line is the point at which the line crosses the x-axis, which means that the y-coordinate at this point is zero. To find the x-intercept from an equation of a line, you simply set \(y = 0\) and solve for \(x\). In the context of the initial problem, the given x-intercept is \(\left(-\frac{2}{3}, 0\right)\).
Finding the x-intercept can help graph a line or understand the behavior of a linear relationship in a real-world context. For example, in a business scenario, an x-intercept might represent a break-even point where no profit or loss is experienced since the corresponding y-value representing profit or loss is zero.

Conversely, the y-intercept is where the line crosses the y-axis, and its x-coordinate is zero. It is found by setting \(x = 0\) in the equation of the line. In our example, the y-intercept is provided as \((0, -2)\). This point is crucial because it often represents the starting value of a relationship modeled by the line.
In the equation \(y = 3x - 2\), the number \(-2\) tells us that the line crosses the y-axis at this point. This is particularly useful when dealing with problems in economics, science, and business, such as determining a fixed cost in a financial model or a starting quantity in a scientific experiment.

The slope of a line measures its steepness and direction, and it's calculated as the change in y over the change in x (\(\frac{\Delta y}{\Delta x}\)). To find the slope, you need two points on the line. In the solution provided, the two points used are the x-intercept and the y-intercept. By applying the slope calculation formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\), and substituting the corresponding coordinates of the points, we obtained the slope as 3.
The slope is a key part of the linear equation, and it helps to understand how one variable affects another. A positive slope means that as x increases, y increases, and vice versa for a negative slope. Zero slope means the line is horizontal, and an undefined slope means the line is vertical, which won't cross the y-axis and therefore doesn't have a y-intercept.