91Ó°ÊÓ

In the context of geometry, the "Point of Intersection" refers to the point where two lines meet each other. To find this point, you need to solve the system of equations that represent the lines. This involves finding the values of the variables (typically \(x\) and \(y\)) that satisfy both equations simultaneously.
For example, in our problem, we have two lines\( x + 3y - 1 = 0 \) and \( 3x - y + 1 = 0 \). When these equations are solved as a system, they reveal the coordinates \( (x, y) \) of their intersection point, \( P \).
Follow these simple steps to find a point of intersection:

• Express one variable in terms of the other from one of the equations.
• Substitute this expression into the second equation.
• Solve the resulting equation for the second variable.
• Substitute back to find the first variable.

This results in the coordinates of the intersection point, making it an essential concept for analyzing where lines cross each other.

The "Slope of a Line" is a measure that describes the steepness and the direction of a line. Calculating the slope is crucial when you want to write an equation of a line through two points or analyze linear relationships.
In mathematical terms, the slope \(m\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by the formula: \( m = \frac{y_2 - y_1}{x_2 - x_1} \). This formula tells us how much the \(y\)-coordinate changes for a unit change in the \(x\)-coordinate.
For instance, in calculating the slope of the line passing through \(C(2,2)\) and \(P\left(\frac{-1}{5}, \frac{2}{5}\right)\), we compute:

• Find the difference in the \(y\)-coordinates: \(2 - \frac{2}{5}\)
• Find the difference in the \(x\)-coordinates: \(2 - \frac{-1}{5}\)
• Calculate \( m = \frac{\frac{8}{5}}{\frac{11}{5}} = \frac{8}{11} \).

The slope \(\frac{8}{11}\) informs us about the line's incline, pointing upwards as it moves from left to right.

Understanding how to derive the "Equation of a Line" is foundational in geometry and algebra. When given a point and a slope, or two points, you can determine a line's equation, describing all the points along the line.
The point-slope form of the equation of a line is: \( y - y_1 = m(x - x_1) \), where \((x_1, y_1)\) is a point on the line and \(m\) is the slope. This form is particularly helpful when you already know these values.
In our exercise, by using a centroid point \((2, 2)\) and a point of intersection \(\left(\frac{-1}{5}, \frac{2}{5}\right)\), we found the equation of the line, after establishing the slope: \( y - 2 = \frac{8}{11}(x - 2) \).
This equation can be further simplified or transformed into other forms such as the slope-intercept form \(y = mx + b\) or standard form \(Ax + By + C = 0\), depending on the needs of the problem.

A "System of Linear Equations" consists of two or more linear equations involving the same set of variables. Solving this system means finding the values of these variables that satisfy all equations at once.
This is a powerful tool used to find the specific points where certain conditions are true, such as the point of intersection between lines.
To solve a system of linear equations, follow these steps:

• Use substitution or elimination methods to reduce the number of variables within the equations.
• For substitution, isolate one variable in one equation and substitute into the other equation.
• For elimination, add or subtract equations to eliminate one variable, making it easier to solve for the other.

In our scenario, we applied substitution by expressing \(x\) from one equation and then substituting it into the other to find \(y\). Solving this system gave us the coordinates of the intersection point \((\frac{-1}{5}, \frac{2}{5})\). This method illustrates the capability of systems of equations in resolving points of convergence between algebraic expressions.