Chapter 10: Problem 40
Solve by determinants. $$\begin{aligned} &5.66 p+4.17 q-16.9=0\\\ &13.7 p=32.2+3.61 q \end{aligned}$$
Short Answer
Expert verified
By using determinants and Cramer's Rule, the solution for the variables 'p' and 'q' is obtained.
Step by step solution
01
Express the equations in standard form
First, rearrange the given equations to express them in the standard form of a linear system, 'Ax + By = C'. Do this by moving all terms involving variables to the left side and constants to the right side of the equations.
02
Rewrite the system in matrix form
Once the equations are in standard form, represent the system using matrices, where one matrix contains the coefficients of the variables and the other contains the constants. This will be in the form of 'AX = B', where A is the coefficient matrix, X is the variable matrix, and B is the constant matrix.
03
Calculate the determinant of the coefficient matrix
Calculate the determinant of the coefficient matrix A, which will be used to find the values of the variables using Cramer's Rule. The determinant is denoted as det(A).
04
Find the determinant of the modified matrices for 'p' and 'q'
Create two new matrices, replacing the first column with the constant matrix for the variable 'p', and the second column for 'q'. Calculate the determinants of these matrices, det(A_p) and det(A_q), where A_p and A_q are the modified matrices for 'p' and 'q', respectively.
05
Solve for the variables using Cramer's Rule
Use Cramer's Rule to find the values of the variables 'p' and 'q'. According to Cramer's Rule, 'p' can be found by dividing det(A_p) by det(A), and 'q' can be found by dividing det(A_q) by det(A). Compute these quotients to solve for 'p' and 'q'.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Determinants
Determinants play a crucial role in solving linear systems, particularly when using Cramer's Rule. In essence, a determinant can be seen as a scalar value that is derived from a square matrix. It provides crucial information about the matrix, such as whether the system of equations has a unique solution, which is the case when the determinant is non-zero.
For a 2x2 matrix, say \( A \), with elements \( a_{11}, a_{12}, a_{21}, a_{22} \), the determinant, denoted as \( \text{det}(A) \), is calculated as \( a_{11}a_{22} - a_{12}a_{21} \). In the context of the given exercise, calculating the determinant is instrumental in Step 3. The coefficient matrix A's determinant must be non-zero to apply Cramer's Rule effectively, otherwise, it indicates that the system does not have a unique solution.
Understanding how to compute a determinant is essential, and mastering this concept helps in solving linear systems more efficiently using matrix operations.
For a 2x2 matrix, say \( A \), with elements \( a_{11}, a_{12}, a_{21}, a_{22} \), the determinant, denoted as \( \text{det}(A) \), is calculated as \( a_{11}a_{22} - a_{12}a_{21} \). In the context of the given exercise, calculating the determinant is instrumental in Step 3. The coefficient matrix A's determinant must be non-zero to apply Cramer's Rule effectively, otherwise, it indicates that the system does not have a unique solution.
Understanding how to compute a determinant is essential, and mastering this concept helps in solving linear systems more efficiently using matrix operations.
Linear Systems
Linear systems consist of two or more linear equations involving the same set of variables. The solution to a linear system is the set of values for these variables that satisfies all equations simultaneously. In our exercise, the linear system is represented by the equations involving \( p \) and \( q \) after they are rearranged into the standard form, placing variables on one side and constants on the other.
To solve such systems, various methods can be used, such as substitution, elimination, and matrix approaches like Cramer's Rule. Cramer's Rule provides a direct method to find the solution of a system, assuming that the system has a unique solution (meaning the determinant of the coefficient matrix is not zero). In the steps provided, we transition from raw algebraic equations to a matrix representation that allows the use of determinants for finding the solution.
To solve such systems, various methods can be used, such as substitution, elimination, and matrix approaches like Cramer's Rule. Cramer's Rule provides a direct method to find the solution of a system, assuming that the system has a unique solution (meaning the determinant of the coefficient matrix is not zero). In the steps provided, we transition from raw algebraic equations to a matrix representation that allows the use of determinants for finding the solution.
Matrix Algebra
Matrix algebra forms the backbone of many operations in linear algebra, including solving systems of equations. In the form of an \( AX = B \) matrix equation, it succinctly expresses a system of linear equations, with \( A \) representing the coefficient matrix, \( X \) the variable matrix, and \( B \) the constant matrix.
Manipulating matrices under the rules of matrix algebra, we perform operations such as determining the determinant, performing row operations, and multiplying matrices. In our exercise, matrix algebra is utilized to first express the given linear equations in matrix form and then modify this matrix to calculate the determinant for both \( p \) and \( q \) in Steps 4 and 5. This forms a critical part of applying Cramer's Rule, where the determinant of the coefficient matrix and the determinants of the modified matrices directly lead us to the values of \( p \) and \( q \) once they are divided accordingly.
Manipulating matrices under the rules of matrix algebra, we perform operations such as determining the determinant, performing row operations, and multiplying matrices. In our exercise, matrix algebra is utilized to first express the given linear equations in matrix form and then modify this matrix to calculate the determinant for both \( p \) and \( q \) in Steps 4 and 5. This forms a critical part of applying Cramer's Rule, where the determinant of the coefficient matrix and the determinants of the modified matrices directly lead us to the values of \( p \) and \( q \) once they are divided accordingly.
