Problem 34 - College Algebra and Calculus: An Applied Approach

Question

Use a determinant to find an equation of the line passing through the points. $$ (3,3),(6,3) $$

Step-by-Step Solution

Answer

The equation of the line passing through the points (3,3) and (6,3) is \(y = 0\).

Step 1: Set up the matrix

First, set up a 2x3 matrix with the given points: \[ \begin{vmatrix} x & y & 1 \\ 3 &3 & 1\\ 6 &3&1 \end{vmatrix} \]. This matrix represents the points (3,3) and (6,3) in the form of coordinates \((x_i,y_i)\).

Step 2: Expand the Determinant

Next, expand the determinant of this matrix using the method of minors. The determinant simplifies into \( Ax + By + C = 0\) where A, B, and C are determinants of 2x2 submatrices: \[ \A = \begin{vmatrix} 3 & 1 \ 3 & 1 \end{vmatrix} = 0\], \[ \B = - \begin{vmatrix} 3 & 1 \ 6 & 1 \end{vmatrix} = -3\], \[ \C = \begin{vmatrix} 3 & 3\ 6 & 3 \end{vmatrix} = 0\]. Having zero for 'A' is fine as this indicates a line parallel to x-axis.

Step 3: Formulate the equation of the line

Finally, substitute A, B and C back into the equation to get: \( 0x -3y + 0 = 0\). After simplifying, the equation for the line is \(y = 0\).

Key Concepts

Matrix

Matrices are fundamental tools in mathematics and are widely used to solve linear equations, among other things. A matrix is essentially a rectangular array of numbers or expressions arranged in rows and columns. For instance, a 2x3 matrix contains two rows and three columns.

The given problem sets up a matrix using the coordinates of two points: (3,3) and (6,3). This matrix representation simplifies the manipulation needed to derive an equation of the line.

Matrices help in organizing data and performing operations such as addition, multiplication, and finding determinants, facilitating complex calculations efficiently.

Equation of a line

The equation of a line in two-dimensional space typically takes the form of either the slope-intercept form, expressed as \(y = mx + b\), or a more generalized form \(Ax + By + C = 0\). For the problem at hand, we use the determinant method to find the equation in the latter form.

Knowing the form of the line's equation is crucial as it dictates the line’s position and orientation in space. For lines parallel to the x-axis, like the one in this problem, the equation simplifies to \(y = c\), where \(c\) is a constant indicating the y-value of the line across all x-values.

Minor expansion

Minor expansion is a technique used to calculate determinants, especially for matrices larger than 2x2. In this approach, you select a row or a column to expand upon and sum the products of each element of that row or column by its corresponding minor and cofactor.

In this problem, for the 2x3 matrix given, minor expansion helps compute a necessary part of the process to extract coefficients A, B, and C, which will eventually plug into our line equation. Selecting appropriate minors ensures you capture the correct relationships between the matrix elements.

2x2 submatrices

Submatrices, particularly the 2x2 variety, play a pivotal role when expanding determinants through minors. A submatrix is a smaller matrix derived from a larger one by cutting out certain rows or columns.

For instance, to find part of the determinant value in this problem, the 2x2 submatrices were created by removing the row containing the element and the column perpendicular to it from the original matrix. This was performed systematically to derive the values A, B, and C.

Understanding how to use 2x2 submatrices helps simplify complex matrices and is essential when dealing with matrix operations in linear algebra.