Problem 42 - College Algebra with Modeling and Visualization

Question

If a line passes through the points \(\left(x_{1}, y_{1}\right)\) and \(\left(x_{2}, y_{2}\right),\) then an equation of this line can be found by calculating the determinant. $$ \operatorname{det}\left[\begin{array}{lll} x & y & 1 \\ x_{1} & y_{1} & 1 \\ x_{2} & y_{2} & 1 \end{array}\right]=0 $$ Find the standard form ax \(+b y=c\) of the line passing through the given points. $$ (-1,3) \text { and }(4,2) $$

Step-by-Step Solution

Answer

The standard form of the line is \(x + 5y = 14\).

Step 1: Set Up the Determinant

Begin by setting up the determinant as described in the problem statement. The points are \((-1, 3)\) and \((4, 2)\). Substitute these values into the matrix:\[\operatorname{det}\left[\begin{array}{lll} x & y & 1 \ -1 & 3 & 1 \ 4 & 2 & 1 \end{array}\right] = 0\]

Step 2: Calculate the Determinant

Calculate the determinant of the matrix using the formula for a 3x3 determinant:\[\operatorname{det}=x(3-2)-y(-1-4)+1(-1 \cdot 2 - 3 \cdot 4)\]Simplify the expression:\[\operatorname{det}=x(1) + y(5) + 1(-2-12)\]\[\operatorname{det}=x + 5y - 14 = 0\]

Step 3: Rearrange to Standard Form

Rearrange the equation from Step 2 into the standard form of a line equation, \(ax + by = c\):\[x + 5y = 14\]This equation already matches the form \(ax + by = c\), where \(a = 1\), \(b = 5\), and \(c = 14\).

Key Concepts

Equation of a Line

An equation of a line is a way to algebraically represent a line in a two-dimensional plane. We can describe a line in many forms, such as the slope-intercept form or the standard form. However, in cases where we have points, using the determinant method can provide an elegant and straightforward solution. When you know two points through which a line passes, it is possible to derive the linear equation by plugging these points into a special formulas.
For instance, using the determinant method involves setting up a 3x3 matrix, aimed at capturing the essence of the given points and the generic formula for any point on the line. The specific setup of the determinant ensures that the resulting condition for the matrix, when set to zero, forms the equation of the line. This method connects geometric intuition with algebra, producing a precise linear equation.

Matrix Determinant

The matrix determinant is a special number calculated from a square matrix. It has wide applications, including solving systems of linear equations and finding out if a set of vectors is linearly independent.
The determinant for a 3x3 matrix is computed via a specific formula, involving the subtraction and addition of products of the matrix's elements. For a matrix:

• det = a(ei − fh) − b(di − fg) + c(dh − eg)

This method takes into account all the elements of the matrix systematically. When applied to our problem, the determinant is calculated from a matrix arrangement of the points

• (−1, 3) and (4, 2) with a generic point

on the line (x, y). Solving this determinant ultimately leads us to derive the linear equation that the matrix represents.

Standard Form of a Line

The standard form of a line is one of the many ways to express the equation of a line, and it is represented as \(ax + by = c\). In this form:

• a, b, and c are real numbers
• a and b are not both zero
• a, b, and c often are integers

This format is preferred for its clarity and simplicity, especially when evaluating the properties of the line and during graphing processes.
The calculation results provide an equation \(x + 5y = 14\), which easily fits in the standard form. Here, \(a = 1\), \(b = 5\), and \(c = 14\), making it easy to interpret the coefficients directly. This form keeps the relationship between x and y proportional and easy to analyze for further mathematical operations.