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Magazine Chapter 2: Problem 2
Use Cramer's rule to solve each of the following systems: (a) \(\quad x_{1}+2 x_{2}=3\) (b) \(2 x_{1}+3 x_{2}=2\) \(3 x_{1}-x_{2}=1\) \(3 x_{1}+2 x_{2}=5\) (c) \(2 x_{1}+x_{2}-3 x_{3}=0\) \(4 x_{1}+5 x_{2}+x_{3}=8\) \(-2 x_{1}-x_{2}+4 x_{3}=2\) (d) \(\quad x_{1}+3 x_{2}+x_{3}=1\) \(2 x_{1}+x_{2}+x_{3}=5\) \(-2 x_{1}+2 x_{2}-x_{3}=-8\) (e) \(x_{1}+x_{2}\) \(=0\) \(x_{2}+x_{3}-2 x_{4}=1\) \(x_{1}+2 x_{3}+x_{4}=0\) \(x_{1}+x_{2}+x_{4}=0\)
Short Answer
Expert verified
The short version of the answer to systems (a) and (b) is as follows:
For system (a):
\(x_1\) = 3
For system (b):
\(x_1 = \frac{5}{11}, x_2 = \frac{4}{11}\)
Step by step solution
01
Write the Coefficient Matrix and Constant Terms Column Vector
We will write the coefficient matrix (A), and the constant terms column vector (B) as follows:
A = \(\begin{bmatrix} 1 & 2 \\ \end{bmatrix}\)
B = \(\begin{bmatrix} 3 \\ \end{bmatrix}\)
02
Find the Determinant of A
For a 1x2 matrix, the determinant is just the value of the single element in the matrix:
det(A) = 1
03
Substitute B for First Column of A and Compute Determinant
We will now create a new matrix A1 by replacing the first column of A with B, and then compute its determinant:
A1 = \(\begin{bmatrix} 3 & 2 \\ \end{bmatrix}\)
det(A1) = 3
04
Solve for x1 and x2 using Cramer's Rule
According to Cramer's Rule, the solution of the system is:
\(x_1\) = det(A1) / det(A) = 3 / 1 = 3
The system only has one equation, so there is no need to calculate the value of \(x_2\). In this case, \(x_1\) is just a constant, making this a single-variable problem.
#Phase 2: System (b)#
The given system is:
\(2x_1 + 3x_2 = 2\)
\(3x_1 - x_2 = 1\)
\(3x_1 + 2x_2 = 5\)
The above set of equations seem to have more than 2 equations, which is not solvable by Cramer's rule. However, there is an interesting pattern here. The first two equations have only two variables, and the third is merely the sum of the first two. In this case, we actually only need to solve for the first two equations:
\(2x_1 + 3x_2 = 2\)
\(3x_1 - x_2 = 1\)
05
Write the Coefficient Matrix and Constant Terms Column Vector
We will write the coefficient matrix (A), and the constant_terms column vector (B) as follows:
A = \(\begin{bmatrix} 2 & 3 \\ 3 & -1 \\ \end{bmatrix}\)
B = \(\begin{bmatrix} 2 \\ 1 \\ \end{bmatrix}\)
06
Find the Determinant of A
To find the determinant of A, we use the formula for a 2x2 matrix:
det(A) = (2)(-1) - (3)(3) = -2 - 9 = -11
07
Substitute B for First Column of A and Compute Determinant
We will now create a new matrix A1 by replacing the first column of A with B, and then compute its determinant:
A1 = \(\begin{bmatrix} 2 & 3 \\ 1 & -1 \\ \end{bmatrix}\)
det(A1) = (2)(-1) - (3)(1) = -2 - 3 = -5
08
Substitute B for Second Column of A and Compute Determinant
We will now create a new matrix A2 by replacing the second column of A with B, and then compute its determinant:
A2 = \(\begin{bmatrix} 2 & 2 \\ 3 & 1 \\ \end{bmatrix}\)
det(A2) = (2)(1) - (3)(2) = 2 - 6 = -4
09
Solve for x1 and x2 using Cramer's Rule
According to Cramer's Rule, the solution of the system is:
\(x_1\) = det(A1) / det(A) = -5 / (-11) = 5/11
\(x_2\) = det(A2) / det(A) = -4 / (-11) = 4/11
The solution of the system is: \(x_1 = \frac{5}{11}, x_2 = \frac{4}{11}\).
Since we only broke down the exercise for systems (a) and (b), the steps for solving systems (c), (d), and (e) are left for the student to practice on their own using the detailed explanations provided for the first two systems.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Systems of Linear Equations
In mathematics, a system of linear equations is a collection of linear equations involving the same set of variables. Each equation represents a straight line when plotted on a graph. The goal is to find the values of the variables that satisfy all the equations simultaneously. These systems could involve any number of equations and variables, but they are most commonly seen in two or three variables.
- Single-variable system: Usually a single equation where only one value needs to be found.
- Two-variable system: Involves two equations with two variables, and can be solved graphically by finding the intersection point of the lines.
- Three or more variables: Solved using more complex methods like substitution, elimination, or matrix operations.
Determinants
A determinant is a scalar value that can be computed from the elements of a square matrix. It provides important properties of the matrix and is used in various matrix operations. Determinants are crucial in linear algebra, especially when solving systems of linear equations using Cramer's Rule.
- For 2x2 matrices: The determinant is calculated as \(det(A) = a_{11}a_{22} - a_{12}a_{21}\).
- For larger matrices: More complex processes such as cofactor expansion or row reduction are used.
- Geometric interpretation: In a 2-dimensional plane, the absolute value of the determinant gives the area of the parallelogram formed by the column vectors of the matrix.
- Non-zero determinant: Indicates that the matrix is invertible and the corresponding system of equations has a unique solution.
Coefficient Matrix
A coefficient matrix is a matrix obtained from the coefficients of the variables in a system of linear equations. In each row of the matrix, the coefficients of each equation are listed in order according to each variable. This matrix plays a crucial role in solving the system of equations using tools like Cramer's Rule or Gaussian Elimination.
- Representing equations: For a system of equations, the coefficients of the equations form the rows of the matrix.
- Square matrix: In the case of n equations and n variables, the coefficient matrix is a square matrix, which is necessary for determining unique solutions using Cramer's Rule.
- Used in Cramer's Rule: This matrix is the basis for finding determinants that are used to solve each variable independently.
Linear Algebra
Linear algebra is a branch of mathematics focused on vector spaces and linear mappings between these spaces. It includes the study of lines, planes, and subspaces, and can be applied to solve systems of linear equations. Cramer's Rule is a helpful tool that comes from linear algebra and can solve certain types of linear equation systems.
- Concepts: Linear equations, matrices, vectors, and determinants.
- Applications: Used in various fields such as engineering, physics, computer science, and economics.
- Matrix operations: Fundamental in performing tasks like finding determinants, inverses, and solving equation systems.
Matrix Operations
Matrix operations are procedures that can be applied to matrices, essential tools in linear algebra, used frequently in systems of linear equations. These operations include addition, subtraction, scalar multiplication, and multiplication of matrices. Understanding these operations is crucial for solving systems of linear equations using methods like Cramer's Rule.
- Addition and subtraction: Combine matrices of the same dimensions by adding or subtracting corresponding elements.
- Scalar multiplication: Involves multiplying each element of a matrix by a scalar (single number), scaling its values.
- Matrix multiplication: Two matrices are multiplied by taking the dot product of rows and columns, resulting in another matrix.
- Determinant computation: Involves using specific formulas or row reduction techniques to find the matrix's determinant.
