Question

Show that only those points satisfying Equation (3) can lie on the line \(L\) passing through \(P_{1}\left(x_{1}, y_{1}\right)\) with slope \(m\).

Short answer

The equation of line L passing through \(P_1(x_1, y_1)\) with slope m is \(y - y_1 = m(x - x_1)\). Rewriting this in the form of Equation (3), we get \(-mx + y - y_1 + mx_1 = 0\). Therefore, only points satisfying Equation (3) can lie on the line L passing through \(P_1(x_1, y_1)\) with slope m.

Step-by-step solution

Equate line L using the point-slope form

The point-slope form of a line passing through a point \((x_1, y_1)\) with slope m is given by: \(y - y_1 = m(x - x_1)\) In this case, the line L passes through \((x_1, y_1)\) with slope m. So, the equation of line L is: \(y - y_1 = m(x - x_1)\)

Rewrite the equation in the form of Equation 3

To show that only points satisfying equation (3) lie on line L, we need to rewrite the equation of line L into the same form as equation (3). Let's assume that equation (3) has the form: \(Ax + By + C = 0\) To rewrite the equation of line L in this form, we need to move all the terms to one side: \(y - y_1 - m(x - x_1) = 0\)

Express A, B, and C in terms of given values

Now, let's express A, B, and C in terms of the given values. From the equation we derived above, we have: \(A = -m\) \(B = 1\) \(C = -y_1 + mx_1\) Now, we have the equation of line L in the form of equation (3): \(-mx + y - y_1 + mx_1 = 0\)

Show that only points satisfying Equation (3) can lie on line L

We have expressed the equation of line L in the form equation (3). Now, any point \((x, y)\) that satisfies the equation can be said to lie on line L. If a point doesn't satisfy this equation, it does not lie on line L. Thus, we've shown that only those points satisfying Equation (3) can lie on the line L passing through \(P_1(x_1, y_1)\) with slope m.

Key concepts

Linear Equations

A linear equation is one of the fundamental concepts in algebra, describing a straight line when plotted on a graph. It is the relationship between two variables, typically x and y, where the highest power of either variable is one. The standard form of a linear equation is denoted as \(Ax + By + C = 0\), where A, B, and C represent real numbers, and A and B are not both zero.

When dealing with linear equations, you can encounter them in different formats, such as the point-slope form, which is especially useful when you have a specific point and the slope of the line. In our exercise example, the point-slope form \(y - y_1 = m(x - x_1)\) clearly expresses how a line's slope and a point on the line determine its equation. Transforming this to the standard form is just a matter of algebraic manipulation, moving all terms to one side of the equation.

Understanding how to transition between different forms of linear equations is crucial. It allows you to analyze and interpret the data that the equation represents, such as determining where the line intersects with the axes and whether a certain point lies on the line.

Slope of a Line

The slope of a line is a measure of its steepness and is denoted by the letter \(m\). Algebraically, it is defined as the 'rise over run', describing how much the line goes up (or down) for a given horizontal distance. In the point-slope equation \(y - y_1 = m(x - x_1)\), \(m\) represents the slope, while \(x_1\), \(y_1\) are the coordinates of a known point on the line.

The slope is a critical component in the characterization of a line because it dictates the angle at which the line tilts from the horizontal. A positive slope means the line is ascending, a negative slope indicates it's descending, and a slope of zero implies the line is horizontal. Infinite slope, where the run is zero, indicates a vertical line.

Calculating Slope from Two Points

If you have two points, \(P_1(x_1, y_1)\) and \(P_2(x_2, y_2)\), you can calculate the slope as \(m = \frac{y_2 - y_1}{x_2 - x_1}\). In the context of our exercise, knowing the slope allows us to establish the orientation and angle of the line in the cartesian plane.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and at least one arithmetic operation (addition, subtraction, multiplication, division). In the context of linear equations, these expressions become the building blocks that we can manipulate to form an equation of a line in various formats.

For example, in the point-slope form \(y - y_1 = m(x - x_1)\), \(y - y_1\) and \(m(x - x_1)\) are algebraic expressions. Through algebraic manipulation, you can expand this point-slope form and rearrange it to the standard linear form \(Ax + By + C = 0\). This process typically involves distributing the slope \(m\) across the \(x\) and \(x_1\) terms, and then moving all variables to one side of the equation to arrive at the coefficients A, B, and C.

Mastering algebraic expressions is essential as it provides the tools needed to transition between different forms of linear equations and to solve for variables in contexts of geometry, algebra, and beyond. Through practice and understanding, you can effortlessly switch between equation forms to suit various mathematical problems and interpret the geometric meaning of a line's equation.