Question

Name the following form of the equation of a line: \(y-y_{1}=m\left(x-x_{1}\right)\)

Short answer

The form of the line equation \(y-y_{1}=m(x-x_{1})\) is point-slope form.

Step-by-step solution

Recall the forms of line equations

The line equation can be presented in various forms: slope-intercept form \(y=mx+b\), point-slope form \(y-y_{1}=m(x-x_{1})\), and standard form \(Ax+By=C\).

Compare the given equation to the known forms

The given equation is \(y-y_{1}=m(x-x_{1})\) which is the point-slope form of the line equation. This form is used when the slope of the line and coordinates of a particular point on the line are known.

Identify the form of the given line equation

As the given line equation matches the point-slope form, we can conclude that the form of the line equation \(y-y_{1}=m(x-x_{1})\) is the point-slope form.

Key concepts

Linear Equations

Understanding linear equations is essential for studying algebra and coordinate geometry. A linear equation represents a straight line on a graph, and it can be written in various forms, each providing different insights about the line. For instance, the slope-intercept form, written as \(y = mx + b\), reveals the slope \(m\) and the y-intercept \(b\). The point-slope form \(y - y_1 = m(x - x_1)\), on the other hand, is very useful when we know the slope of the line and one point \( (x_1, y_1)\) through which the line passes.
In coordinate geometry, an understanding of linear equations helps students plot lines and analyze their characteristics. The equation \( Ax + By = C \) is the standard form, where \(A\), \(B\), and \(C\) are integers. This form can be particularly helpful when solving systems of linear equations or finding the x- and y-intercepts of a line.
To delve deeper into the concept, let's explore the importance of slope and how it relates to the point-slope form in the next section.

Slope of a Line

The slope of a line is a measure of its steepness and direction. Mathematically, it's the ratio of the rise over the run between any two points on the line. In the equation \(y - y_1 = m(x - x_1)\), \(m\) represents the line's slope. For positive slopes, the line ascends from left to right, while a negative slope indicates the line descends. A horizontal line has a slope of zero, and a vertical line has an undefined slope.
When dealing with the point-slope form, the slope is a critical piece of information, as it allows you to determine the direction and angle of the line with respect to the x-axis. This concept is used in many applications, such as calculating inclines, predicting future values in data analysis, and designing roof slopes in architecture.

Connecting Slope to the Point-Slope Form

To connect slope to the point-slope form, identify a known point on the line. Using the slope, you can find other points or write an equation for the line that passes through the known point by substituting the slope and the coordinates of the known point into the point-slope form.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, involves the study of geometric figures using the coordinate plane. This branch of geometry uses algebraic equations to represent geometric shapes and can often simplify problems that are difficult to solve using classical geometric methods.
In coordinate geometry, each point is defined by a pair of numerical coordinates, which are the distances from the point to two fixed perpendicular lines, known as axes. For example, \(y - y_1 = m(x - x_1)\) uses the coordinates \( (x_1, y_1)\) to define a specific point on the line, and \(m\) to describe its slope.
This unification of algebra and geometry allows for analyzing and solving problems concerning lengths, areas, and slopes. Understanding the concept of coordinate geometry is fundamental when learning about the point-slope form and other forms of linear equations, as it forms the basis for plotting and visualizing these equations on a graph.