Question

In Exercises \(13-16,\) write the point-slope equation for the line through the point \(P\) with slope \(m .\) $$P(1,1), \quad m=1$$

Short answer

The point-slope equation of the line is \(y - 1 = x - 1\) or equivalently \(y - x = 0\).

Step-by-step solution

Identify and label the required components

From the exercise, point \(P\) is given as \((1,1)\), so \(x_1 = 1\) and \(y_1 = 1\). The slope \(m\) is given as \(1\).

Substitute the values into the point-slope equation

Now, substitute \(x_1, y_1\) and \(m\) into the point-slope formula \(y - y_1 = m(x - x_1)\). On substitution, we have \(y - 1 = 1*(x - 1)\).

Simplify the substitution

After the substitution, the equation becomes \(y - 1 = x - 1\). Since the question only asks for the point-slope form, no further simplification is required. However, if needed, it can be reformatted to: \(y - x = 0\).

Key concepts

Slope-Intercept Form

The slope-intercept form is an equation of a straight line in the format of \(y = mx + b\), where \(m\) represents the slope of the line and \(b\) indicates the y-intercept, which is the point where the line crosses the y-axis. This form is particularly useful as it allows you to quickly identify both the slope and the y-intercept from the equation itself.

For instance, in the given exercise, after simplifying the point-slope equation \(y - 1 = x - 1\), if one were to convert it into the slope-intercept form, they would add 1 to each side of the equation resulting in \(y = x\). This simple representation immediately gives you the information that the slope \(m\) is 1, and the y-intercept \(b\) is 0, which means the line crosses the y-axis at the origin.

Linear Equations

Linear equations represent straight lines and are fundamental in algebra. A linear equation can be expressed in various forms, each providing different insights into the line's properties. The general form is \(Ax + By + C = 0\), where \(A\), \(B\), and \(C\) are constants. The slope-intercept form and the point-slope form are specific types of linear equations.

Understanding the relationship between these forms is crucial. For example, the point-slope form \(y - y_1 = m(x - x_1)\) derived in the solution can be modified to the general form or converted to the slope-intercept form, depending on what information you need or what properties of the line you are interested in.

Algebraic Representation

Algebraic representation involves expressing mathematical concepts and relationships using symbols and letters, such as the equation we derived in the problem. This symbolic language allows us to generalize mathematical ideas and manipulate equations more efficiently.

In the context of the given exercise, we utilized algebraic representation to formulate the point-slope equation based on known values of a point and the slope. This representation is powerful because it not only conveys the particular solution but also sets the stage for further algebraic manipulation if necessary.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is the study of geometry using a coordinate system. This field links algebraic equations to geometric curves and figures by placing them on a grid defined by an x and y axis.

The point-slope equation from the exercise is a perfect example of how coordinate geometry combines both algebra and geometry. The equation \(y - y_1 = m(x - x_1)\) represents a line that passes through a specific point \(P\) and has a certain slope \(m\). This connects an algebraic formula with a geometric line on a plane, thus allowing for a deep understanding and analysis of geometric figures through algebraic methods.