Problem 3 - Precalculus

Question

The_______ , ______ - form of the equation of a line with slope \(m\) passing through \(\left(x_{1}, y_{1}\right)\) is \(y-y_{1}=m\left(x-x_{1}\right)\).

Step-by-Step Solution

Answer

The slope-point form of equation of a line is \(y - y_{1} = m(x - x_{1})\). In this form, \(m\) represents the slope of the line and \((x_{1}, y_{1})\) represent any point through which the line passes.

Step 1: Understand the given form

Look at the given equation \(y - y_{1} = m(x - x_{1})\). The equation represents the slope-point form of a line. Here, \(m\) is the slope of the line, and \((x_{1}, y_{1})\) is the point through which the line passes.

Step 2: Derivation of slope-point form

To derive this form, start with the formula for the slope of line which is \(m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}\). Here, \((x_{1}, y_{1})\) and \((x_{2}, y_{2})\) are two points on the line. Now, take any general point on the line as \((x, y)\) which replaces \((x_{2}, y_{2})\) in slope formula resulting in \(m = \frac{y - y_{1}}{x - x_{1}}\). Rewriting this gives \(y - y_{1} = m(x - x_{1})\). This is the slope-point form of a line.

Step 3: Use of slope-point form

This form of the equation is used when the slope of the line and a point through which the line passes are given. It provides a link between the slope of a line and the coordinates of a specific point on the line.

Key Concepts

Line Equation

The line equation is a fundamental concept in algebra and coordinate geometry that helps describe a line using a mathematical formula. One of the most common forms for representing a line is the slope-point form, given by the equation:

• \(y - y_{1} = m(x - x_{1})\)

In this equation:

• \(m\) represents the slope of the line.
• \((x_{1}, y_{1})\) is a point on the line.

This form is particularly useful when you know a point through which the line passes along with the slope. It provides a direct way to construct the equation of a line using these values. The slope-point form is a powerful tool for transforming the abstract idea of a line into a concrete mathematical expression capable of describing the line's behavior throughout the coordinate plane.

Slope

The slope of a line is a key concept in understanding how the line behaves. It measures the steepness or incline of the line and is typically represented by the letter \(m\). The slope is defined mathematically as the ratio of the vertical change to the horizontal change between two points on the line. This can be expressed as:

• \(m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}\)

Where:

• \((x_{1}, y_{1})\) and \((x_{2}, y_{2})\) are two distinct points on the line.

The slope tells us how many units the line rises or falls for each unit it moves horizontally. A positive slope indicates the line rises as it moves from left to right, while a negative slope shows it falls. A slope of zero signifies a horizontal line, and an undefined slope corresponds to a vertical line. Understanding slope is crucial for analyzing and predicting the direction of lines in coordinate geometry and for solving more complex geometric problems.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is a branch of mathematics that allows geometric problems to be solved using algebraic equations. It involves representing geometric figures and analyzing their properties using coordinates on a Cartesian plane. In this system:

• Every point is identified by an ordered pair \((x, y)\).
• The horizontal axis is the x-axis, and the vertical axis is the y-axis.

Coordinate geometry provides a comprehensive framework for solving problems involving shapes, sizes, and the relative positions of figures in a plane. By utilizing equations like the slope-point form \(y - y_{1} = m(x - x_{1})\), it becomes possible to model and solve real-world problems mathematically. This powerful combination of geometry and algebra helps to better understand the spatial relationships between different figures and to calculate distances, angles, and areas with precision. Understanding coordinate geometry is fundamental to both algebra and geometry, providing insightful connections between algebraic formulas and geometric figures.