91Ó°ÊÓ

The slope of a line is a measure of how steep the line is, and it indicates the relationship between the change in the vertical direction (rise) and the change in the horizontal direction (run) between two points on the line. Mathematically, the slope is defined as:\[m = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1}\]where

• \((x_1, y_1)\) and \((x_2, y_2)\) are coordinates of two distinct points on the line,
• \(\Delta y\) is the change in the y-coordinates, and
• \(\Delta x\) is the change in the x-coordinates.

The slope tells us many things about a line:

• If \(m > 0\), the line is sloping upwards from left to right.
• If \(m < 0\), the line is sloping downwards.
• If \(m = 0\), the line is horizontal, and
• If the line is vertical, the slope is undefined because \(\Delta x = 0\).

In the context of the given equation, the slope \(m\) is shown through the process of expanding the determinant. This indicates the line's inclination compared to a horizontal line.

The point-slope form of a line is a handy way to describe a straight line when you know one point on the line and its slope. The equation is given by:\[y - y_1 = m(x - x_1)\]where

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

This form is especially useful because it directly incorporates the slope and a specific point, making it easier to visualize and understand the line's behavior.
In our exercise, after expanding and simplifying the determinant, we end up with this same form, illustrating that the expression represents a line through the given point with the specified slope. This shows how determinants can be used to derive the equation of a line efficiently, by simplifying geometric properties into algebraic expressions.

Determinants offer a powerful method to handle equations in matrix form. By expanding determinants, as in this exercise, we reformulate a matrix into a more comprehensible algebraic expression.
The process of expanding the determinant \(\left|\begin{array}{lll}x & y & 1 \ x_{1} & y_{1} & 1 \ 1 & m & 0\end{array}\right|\) involves computing the product of its diagonal terms and then subtracting the product of other minor matrices. This results in:\[x(y_1 \cdot 0) - xm - y(x_1 \cdot 0 + 1) + 1(x_1 \cdot m - y_1 \cdot 1)\]through careful expansion of each row.
Upon simplification, this expression becomes \(y - y_1 = m(x - x_1)\), which, as we discovered, describes the point-slope form of a line.

• Expanding determinants can simplify complex matrix expressions into a form that is easier to analyze.
• This technique helps relate the properties of the matrix (or equation) to geometric concepts such as lines and their slopes.

Thus, by understanding determinants and their expansions, you gain a useful tool for translating geometric scenarios into algebraic language.