What Is The Solution To This System Of Linear Equations

Solving a system of linear equations is a fundamental concept in mathematics with broad applications in various fields, including engineering, physics, economics, and computer science. Understanding the methods to find the solution to such a system is crucial for anyone involved in quantitative analysis and problem-solving.

Introduction to Systems of Linear Equations

A system of linear equations comprises two or more linear equations that share the same variables. The solution to a system of linear equations is the set of values for the variables that satisfy all equations simultaneously. This means that when these values are substituted into each equation, the left-hand side equals the right-hand side.

Forms of Linear Equations

A linear equation can generally be written in the form:

a₁x₁ + a₂x₂ + ... + aₙxₙ = b

Where:

x₁, x₂, ..., xₙ are the variables.
a₁, a₂, ..., aₙ are the coefficients of the variables.
b is the constant term.

Types of Solutions

A system of linear equations can have one of three types of solutions:

Unique Solution: There is exactly one set of values for the variables that satisfies all equations.
No Solution: There is no set of values for the variables that can satisfy all equations simultaneously. The system is inconsistent.
Infinitely Many Solutions: There are an infinite number of sets of values for the variables that satisfy all equations. This typically occurs when the equations are dependent.

Methods to Solve Systems of Linear Equations

Several methods are available to solve systems of linear equations, each with its own advantages and complexities. The choice of method depends on the number of equations, the number of variables, and the structure of the equations. Here, we'll discuss some of the most common methods:

Substitution Method
Elimination Method (also known as the Addition Method)
Gaussian Elimination
Matrix Methods (including Inverse Matrix and Cramer's Rule)
Graphical Method

1. Substitution Method

The substitution method involves solving one equation for one variable and then substituting that expression into another equation. This process reduces the system to a single equation with one variable, which can then be easily solved.

Steps:

Solve one equation for one variable: Choose one equation and solve it for one of the variables. This will express one variable in terms of the others.
Substitute: Substitute the expression obtained in step 1 into another equation. This will eliminate one variable from the second equation.
Solve the resulting equation: Solve the equation obtained in step 2 for the remaining variable.
Back-substitute: Substitute the value obtained in step 3 back into the expression from step 1 to find the value of the other variable.
Check the solution: Substitute the values of all variables into the original equations to ensure they are satisfied.

Example:

Consider the system of equations:

x + y = 5
2x - y = 1

Solution:

Solve for x in the first equation: x = 5 - y
Substitute into the second equation: 2(5 - y) - y = 1 10 - 2y - y = 1 10 - 3y = 1
Solve for y: -3y = -9 y = 3
Back-substitute to find x: x = 5 - 3 x = 2
Check the solution: 2 + 3 = 5 (Correct) 2(2) - 3 = 1 (Correct)

The solution is x = 2 and y = 3.

2. Elimination Method

The elimination method (or addition method) involves adding or subtracting multiples of the equations to eliminate one of the variables. The goal is to manipulate the equations so that the coefficients of one variable are opposites, allowing them to cancel out when the equations are added.

Steps:

Align the equations: Write the equations so that like terms are aligned in columns.
Multiply equations: Multiply one or both equations by constants so that the coefficients of one variable are opposites (or the same).
Add or subtract equations: Add (or subtract) the equations to eliminate one variable.
Solve the resulting equation: Solve the equation obtained in step 3 for the remaining variable.
Back-substitute: Substitute the value obtained in step 4 back into one of the original equations to find the value of the other variable.
Check the solution: Substitute the values of all variables into the original equations to ensure they are satisfied.

Example:

Consider the system of equations:

3x + 2y = 7
x - y = -1

Solution:

Align the equations: 3x + 2y = 7 x - y = -1
Multiply the second equation by 2: 3x + 2y = 7 2x - 2y = -2
Add the equations: (3x + 2y) + (2x - 2y) = 7 + (-2) 5x = 5
Solve for x: x = 1
Back-substitute into the second original equation: 1 - y = -1 -y = -2 y = 2
Check the solution: 3(1) + 2(2) = 3 + 4 = 7 (Correct) 1 - 2 = -1 (Correct)

The solution is x = 1 and y = 2.

3. Gaussian Elimination

Gaussian elimination is a systematic method used to solve systems of linear equations by transforming the system into an upper triangular form using elementary row operations. This method is particularly useful for larger systems of equations.

Steps:

Write the augmented matrix: Represent the system of equations as an augmented matrix. This matrix consists of the coefficients of the variables and the constants, separated by a vertical line.
Perform row operations to get the matrix in row-echelon form: Use elementary row operations to transform the matrix into row-echelon form. Elementary row operations include:
- Swapping two rows.
- Multiplying a row by a non-zero constant.
- Adding a multiple of one row to another row.
Perform row operations to get the matrix in reduced row-echelon form: Continue using elementary row operations to transform the matrix into reduced row-echelon form. In this form, the leading entry (pivot) in each row is 1, and all other entries in the column containing a pivot are 0.
Solve for the variables: Read the solution directly from the reduced row-echelon form of the matrix.

Example:

Consider the system of equations:

x + y + z = 6
2x - y + z = 3
x + 2y - z = 2

Solution:

Write the augmented matrix:

[ 1  1  1 | 6 ]
[ 2 -1  1 | 3 ]
[ 1  2 -1 | 2 ]

Perform row operations to get the matrix in row-echelon form:

R₂ → R₂ - 2R₁:

[ 1  1  1 | 6 ]
[ 0 -3 -1 | -9 ]
[ 1  2 -1 | 2 ]

R₃ → R₃ - R₁:

[ 1  1  1 | 6 ]
[ 0 -3 -1 | -9 ]
[ 0  1 -2 | -4 ]

Swap R₂ and R₃:

[ 1  1  1 | 6 ]
[ 0  1 -2 | -4 ]
[ 0 -3 -1 | -9 ]

R₃ → R₃ + 3R₂:

[ 1  1  1 | 6 ]
[ 0  1 -2 | -4 ]
[ 0  0 -7 | -21 ]

Perform row operations to get the matrix in reduced row-echelon form:

R₃ → R₃ / -7:

[ 1  1  1 | 6 ]
[ 0  1 -2 | -4 ]
[ 0  0  1 | 3 ]

R₂ → R₂ + 2R₃:

[ 1  1  1 | 6 ]
[ 0  1  0 | 2 ]
[ 0  0  1 | 3 ]

R₁ → R₁ - R₃:

[ 1  1  0 | 3 ]
[ 0  1  0 | 2 ]
[ 0  0  1 | 3 ]

R₁ → R₁ - R₂:

[ 1  0  0 | 1 ]
[ 0  1  0 | 2 ]
[ 0  0  1 | 3 ]

Solve for the variables:
- x = 1
- y = 2
- z = 3

The solution is x = 1, y = 2, and z = 3.

4. Matrix Methods

Matrix methods are powerful tools for solving systems of linear equations, particularly when dealing with larger systems. Two common matrix methods are the inverse matrix method and Cramer's rule.

a. Inverse Matrix Method

The inverse matrix method involves expressing the system of linear equations in matrix form and then solving for the variables by multiplying both sides of the equation by the inverse of the coefficient matrix.

Steps:

Write the system in matrix form: Express the system of equations in the form AX = B, where A is the coefficient matrix, X is the column matrix of variables, and B is the column matrix of constants.
Find the inverse of the coefficient matrix: Calculate the inverse of matrix A, denoted as A⁻¹. The inverse exists only if the determinant of A is non-zero.
Solve for the variables: Multiply both sides of the equation AX = B by A⁻¹ to obtain X = A⁻¹B. Compute the matrix product A⁻¹B to find the values of the variables.

Example:

Consider the system of equations:

2x + y = 7
x - y = -1

Solution:

Write the system in matrix form:

A = [ 2  1 ]
    [ 1 -1 ]

X = [ x ]
    [ y ]

B = [ 7 ]
    [ -1 ]

AX = B is:

[ 2  1 ] [ x ] = [ 7 ]
[ 1 -1 ] [ y ] = [ -1 ]

Find the inverse of the coefficient matrix: The determinant of A is (2 * -1) - (1 * 1) = -2 - 1 = -3. A⁻¹ = (-1/3) * [-1 -1] [-1 2] = [ 1/3 1/3] [ 1/3 -2/3]

Solve for the variables: X = A⁻¹B

[ x ] = [ 1/3   1/3] [ 7 ]
[ y ] = [ 1/3  -2/3] [ -1]

[ x ] = [ (1/3 * 7) + (1/3 * -1) ] = [ 6/3 ] = [ 2 ]
[ y ] = [ (1/3 * 7) + (-2/3 * -1) ] = [ 9/3 ] = [ 3 ]

The solution is x = 2 and y = 3.

b. Cramer's Rule

Cramer's rule is a method for solving systems of linear equations using determinants. It provides a formula for finding the value of each variable in terms of determinants of matrices derived from the coefficient matrix and the constant matrix.

Steps:

Write the system in matrix form: Express the system of equations in the form AX = B, where A is the coefficient matrix, X is the column matrix of variables, and B is the column matrix of constants.
Calculate the determinant of the coefficient matrix: Find the determinant of matrix A, denoted as det(A).
Calculate the determinants for each variable: For each variable xᵢ, replace the i-th column of matrix A with the constant matrix B to form a new matrix Aᵢ. Calculate the determinant of each Aᵢ.
Solve for the variables: Use the formula xᵢ = det(Aᵢ) / det(A) to find the value of each variable.

Example:

Consider the system of equations:

2x + y = 7
x - y = -1

Solution:

Write the system in matrix form:

A = [ 2  1 ]
    [ 1 -1 ]

X = [ x ]
    [ y ]

B = [ 7 ]
    [ -1 ]

Calculate the determinant of the coefficient matrix: det(A) = (2 * -1) - (1 * 1) = -2 - 1 = -3.
Calculate the determinants for each variable: For x:
```
Ax = [ 7  1 ]
     [ -1 -1 ]
```
det(Ax) = (7 * -1) - (1 * -1) = -7 + 1 = -6. For y:
```
Ay = [ 2  7 ]
     [ 1 -1 ]
```
det(Ay) = (2 * -1) - (7 * 1) = -2 - 7 = -9.
Solve for the variables: x = det(Ax) / det(A) = -6 / -3 = 2 y = det(Ay) / det(A) = -9 / -3 = 3

The solution is x = 2 and y = 3.

5. Graphical Method

The graphical method is used to solve systems of linear equations by plotting the equations on a coordinate plane and finding the point of intersection. This method is primarily used for systems of two equations with two variables.

Steps:

Rewrite the equations in slope-intercept form: Convert each equation into the form y = mx + b, where m is the slope and b is the y-intercept.
Plot the lines: Draw the lines corresponding to each equation on a coordinate plane.
Find the point of intersection: Identify the point where the lines intersect. The coordinates of this point represent the solution to the system of equations.
Verify the solution: Substitute the coordinates of the intersection point into the original equations to ensure they are satisfied.

Example:

Consider the system of equations:

x + y = 5
2x - y = 1

Solution:

Rewrite the equations in slope-intercept form:
- y = -x + 5
- y = 2x - 1
Plot the lines: Draw the lines y = -x + 5 and y = 2x - 1 on a coordinate plane.
Find the point of intersection: The lines intersect at the point (2, 3).
Verify the solution:
- 2 + 3 = 5 (Correct)
- 2(2) - 3 = 1 (Correct)

The solution is x = 2 and y = 3.

Special Cases

No Solution

When a system of linear equations has no solution, it is said to be inconsistent. Graphically, this occurs when the lines are parallel and do not intersect. Algebraically, this is indicated when the equations lead to a contradiction.

Example:

Consider the system of equations:

x + y = 3
x + y = 5

Subtracting the first equation from the second, we get:

0 = 2, which is a contradiction. Therefore, there is no solution.

Infinitely Many Solutions

When a system of linear equations has infinitely many solutions, the equations are dependent. Graphically, this occurs when the lines are coincident (i.e., they are the same line). Algebraically, this is indicated when one equation is a multiple of another.

Example:

Consider the system of equations:

x + y = 3
2x + 2y = 6

The second equation is simply twice the first equation. Therefore, the system has infinitely many solutions.

Applications of Systems of Linear Equations

Systems of linear equations are used to model and solve a wide range of problems in various fields:

Engineering: Solving for currents and voltages in electrical circuits, analyzing forces in structures, and designing control systems.
Physics: Modeling motion, solving for forces and accelerations, and analyzing thermodynamic systems.
Economics: Determining equilibrium prices and quantities in markets, analyzing economic models, and optimizing resource allocation.
Computer Science: Solving linear programming problems, performing data analysis, and creating computer graphics.
Statistics: Performing regression analysis, solving for parameters in statistical models, and analyzing data sets.

Conclusion

Solving systems of linear equations is a fundamental skill in mathematics and has broad applications across various disciplines. The methods discussed—substitution, elimination, Gaussian elimination, matrix methods (inverse matrix and Cramer's rule), and the graphical method—provide a comprehensive toolkit for tackling different types of systems. Understanding the nuances of each method, as well as recognizing special cases such as systems with no solution or infinitely many solutions, is crucial for effective problem-solving. Whether you're an engineer, scientist, economist, or computer scientist, mastering the techniques for solving systems of linear equations will undoubtedly enhance your ability to analyze and solve complex problems.

What Is The Solution To This System Of Linear Equations

Table of Contents

Introduction to Systems of Linear Equations

Forms of Linear Equations

Types of Solutions

Methods to Solve Systems of Linear Equations

1. Substitution Method

2. Elimination Method

3. Gaussian Elimination

4. Matrix Methods

a. Inverse Matrix Method

b. Cramer's Rule

5. Graphical Method

Special Cases

No Solution

Infinitely Many Solutions

Applications of Systems of Linear Equations

Conclusion

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