91Ó°ÊÓ

Understanding the equation of a line is crucial for various applications in geometry and algebra. The equation represents a straight path on a two-dimensional graph. The most commonly used forms of a line's equation are the slope-intercept form (\(y = mx + b\)), point-slope form (\(y - y_1 = m(x - x_1)\) where \(m\) is the slope, and \(x_1\text{, } y_1\text{,}\text{ and } b\text{ }\text{are constants.}\text{)}\), and standard form (\(Ax + By = C\text{,}\text{ where } A\text{, } B\text{, and } C \text{ are integers.}\text{)}\). The point-slope form is especially useful when you know a point on the line \(\left(x_1, y_1\right)\) and the slope. This form is particularly expedient for sketching the graph of a line quickly and for understanding its behavior.For example, if you have the points \(2, 1\right)\) and \(3, 4\right)\), and you want to find the equation of the line passing through them, you can calculate the slope and use one of the points to apply the point-slope form. In various problems, like the ones mentioned in the exercise, this form facilitates the construction of the line's equation with the given information.

The slope of a line measures its steepness and direction. It is a ratio that describes how much the line rises or falls for every unit of horizontal movement. Mathematically, the slope \(m\) between two points \(\left(x_1, y_1\right)\) and \(\left(x_2, y_2\right)\) is calculated using the formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\).

• If the slope is positive, the line rises from left to right.
• If the slope is negative, the line falls from left to right.
• If the slope is zero, the line is horizontal.
• If the slope is undefined (because of a division by zero), the line is vertical.

In the given problems, you can see how this computation allows us to identify the direction and angle of the lines in question. For instance, the slope of the line passing through \(\left(2, 1\right)\) and \(\left(3, 4\right)\) is 3, revealing that for every step rightward, the line ascends by three units.

The x-intercept of a line refers to the point where the line crosses the x-axis. This point is of the form \(\left(x, 0\right)\) since the y-coordinate is zero whenever a line intersects the x-axis. To find the x-intercept mathematically, one typically sets the y-component of the equation to zero and solves for x. It is an important characteristic, helping to define a line's equation and providing a valuable starting point for graphing.When given a line's slope and an x-intercept, as in the problems with the lines that have x-intercepts of 2 and 3, the point-slope form becomes a useful tool for formulating the line's equation. Using the x-intercept and applying the slope, it becomes straightforward to derive the rest of the line's features.

In geometry, perpendicular lines are lines that intersect at a right angle (90 degrees). The slopes of two perpendicular lines are negative reciprocals of one another. To find the slope of a line perpendicular to another, you take the negative reciprocal of the given line's slope. If a line has a slope \(m\), the slope of a line perpendicular to it will be \(\frac{-1}{m}\) or \(\frac{-1}{m}\).In our exercise, the given line has an equation of \(3y = -2x + 24\), with a slope of \(m = -\frac{2}{3}\). Therefore, the slope of a line perpendicular to it is \(\frac{3}{2}\). This concept is fundamental when solving problems involving perpendicular lines, as it ensures the two lines will indeed intersect at that crucial right angle. The process demonstrates the beauty of the interconnectedness in mathematics—where a single property like the slope can tell so much about the spatial relationship between two lines.