Problem 36 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 33-42: Use Cramer's rule, whenever applicable, to solve the system. $$ \left\\{\begin{array}{l} 7 x-8 y=9 \\ 4 x+3 y=-10 \end{array}\right. $$

Step-by-Step Solution

Answer

The solution is \( x = -1 \) and \( y = -2 \).

Step 1: Calculate the Determinant of the Coefficient Matrix

First, we write the coefficient matrix of the system which is \[A = \begin{bmatrix} 7 & -8 \ 4 & 3 \end{bmatrix}\] The determinant of this matrix is computed as follows:\[det(A) = (7)(3) - (-8)(4) = 21 + 32 = 53\]Since \( det(A) eq 0 \), Cramer's Rule is applicable.

Step 2: Compute Determinant for X

To find the determinant for \( x \), replace the first column of the coefficient matrix \( A \) with the constants from the equations:\[A_x = \begin{bmatrix} 9 & -8 \ -10 & 3 \end{bmatrix}\]Calculate \( det(A_x) \):\[det(A_x) = (9)(3) - (-8)(-10) = 27 - 80 = -53\]

Step 3: Compute Determinant for Y

To find the determinant for \( y \), replace the second column of the coefficient matrix \( A \) with the constants:\[A_y = \begin{bmatrix} 7 & 9 \ 4 & -10 \end{bmatrix}\]Calculate \( det(A_y) \):\[det(A_y) = (7)(-10) - (9)(4) = -70 - 36 = -106\]

Step 4: Solve for X and Y Using Cramer's Rule

Using Cramer's Rule, solve for \( x \) and \( y \):\[x = \frac{det(A_x)}{det(A)} = \frac{-53}{53} = -1\]\[y = \frac{det(A_y)}{det(A)} = \frac{-106}{53} = -2\]

Key Concepts

Determinant

The determinant is a special number that can be calculated from a square matrix. It is fundamental in finding solutions to systems of linear equations and plays a key role in linear algebra. The value of the determinant gives us important insights about a matrix, such as whether it is invertible.

To determine the determinant of a 2x2 matrix:

• Multiply the elements in the main diagonal.
• Subtract the product of the elements in the opposite diagonal.

For example, given the matrix \[A = \begin{bmatrix} a & b \ c & d \end{bmatrix},\]the determinant is \[ad - bc.\]

If the determinant is zero, the matrix does not have an inverse, and systems of equations involving such a matrix may not have a unique solution.

System of Equations

A system of equations consists of two or more equations with the same set of unknowns. Solving a system of equations means finding values for these unknowns that satisfy every equation in the system simultaneously.

Systems arise in various fields such as physics and economics, where multiple conditions or constraints need to be met at once. For instance, given the system:\[\left\{\begin{array}{l}7x - 8y = 9 \4x + 3y = -10\end{array}\right.\]our aim is to find the values of \(x\) and \(y\) that make both equations true.

Different methods exist to solve these systems, including substitution, elimination, and using matrices like in Cramer's Rule.

Matrices

Matrices are arrays of numbers arranged in rows and columns. They are used extensively in mathematics to represent systems of linear equations, transformations, and more.

A matrix can be of any size, but in this context, we often refer to relatively small ones, such as 2x2 or 3x3. Each matrix has elements denoted by the position in the row and column.

• Rows refer to the horizontal lines of elements in a matrix.
• Columns are the vertical lines of elements.

Matrices can be added, subtracted, and multiplied under certain rules. Furthermore, they can be used to represent and solve systems of linear equations through techniques like Cramer's Rule. Understanding matrices is fundamental in linear algebra because they allow us to perform complex calculations in a structured way.

Linear Algebra

Linear algebra is a branch of mathematics that focuses on vector spaces and linear mappings between them. It encompasses the study of lines, planes, and subspaces, but more importantly, it also explores systems of linear equations and the properties of matrices.

This subject is foundational in various domains including engineering, computer science, physics, and more. It provides tools that are crucial for performing operations on matrices, such as finding determinants, solving matrix equations, and understanding linear transformations.

• Linear algebra helps to simplify complex problems by converting them into manageable, numerical forms.
• It allows the formulation of mathematical models to solve real-world problems efficiently.

Cramer's Rule is one such technique derived from linear algebra principles, which utilizes determinants of matrices to solve systems of linear equations.