Undergraduate
  1. Algebra  1.1. Linear Algebra  1.1.1. Matrices  1.1.2. Determinants  1.1.3. Systems of Linear Equations  1.1.4. Vector Spaces  1.1.5. Linear Transformations  1.1.6. Eigenvalues and Eigenvectors  1.1.7. Diagonalization  1.1.8. Inner Product Spaces  1.1.9. Orthogonality and Orthonormal Bases  1.1.10. Singular Value Decomposition  1.2. Abstract Algebra  1.2.1. Groups  1.2.2. Subgroups  1.2.3. Cyclic Groups  1.2.4. Permutation Groups  1.2.5. Homomorphisms  1.2.6. Normal Subgroups  1.2.7. Rings  1.2.8. Fields  1.2.9. Ideals and Quotient Rings  1.2.10. Polynomial Rings  1.2.11. Vector Spaces over Fields  1.2.12. Modules  1.2.13. Galois Theory
  2. Calculus  2.1. Differential Calculus  2.1.1. Limits  2.1.2. Continuity  2.1.3. Derivatives  2.1.4. Chain Rule  2.1.5. Implicit Differentiation  2.1.6. Applications of Derivatives  2.1.7. Optimization Problems  2.1.8. Mean Value Theorem  2.2. Integral Calculus  2.2.1. Definite Integrals  2.2.2. Indefinite Integrals  2.2.3. Techniques of Integration  2.2.4. Improper Integrals  2.2.5. Applications of Integration  2.2.6. Arc Length and Surface Area  2.3. Multivariable Calculus  2.3.1. Partial Derivatives  2.3.2. Gradient  2.3.3. Divergence and Curl  2.3.4. Multiple Integrals  2.3.5. Line Integrals  2.3.6. Surface Integrals  2.3.7. Green's Theorem  2.3.8. Stokes' Theorem  2.3.9. Divergence Theorem  2.3.10. Jacobians
  3. Differential Equations  3.1. Ordinary Differential Equations  3.1.1. First Order Differential Equations  3.1.2. Higher Order Differential Equations  3.1.3. Systems of Differential Equations  3.1.4. Series Solutions  3.1.5. Numerical Methods for ODEs  3.2. Partial Differential Equations  3.2.1. Heat Equation  3.2.2. Wave Equation  3.2.3. Laplace Equation  3.2.4. Separation of Variables  3.2.5. Fourier Transform Methods
  4. Real Analysis  4.1. Sequences and Series  4.1.1. Convergence of Sequences  4.1.2. Series Tests  4.1.3. Power Series  4.1.4. Uniform Convergence  4.2. Functions of Real Variables  4.2.1. Limits and Continuity  4.2.2. Uniform Continuity  4.2.3. Differentiation in Functions of Real Variables  4.2.4. Riemann Integration  4.2.5. Lebesgue Integration
  5. Complex Analysis  5.1. Complex Numbers  5.1.1. Polar Form  5.1.2. Exponential Form  5.2. Functions of a Complex Variable  5.2.1. Analytic Functions  5.2.2. Cauchy-Riemann Equations  5.2.3. Contour Integrals  5.2.4. Cauchy's Integral Theorem  5.2.5. Residue Theorem  5.2.6. Conformal Mapping
  6. Probability and Statistics  6.1. Probability Theory  6.1.1. Basic Probability Concepts  6.1.2. Random Variables  6.1.3. Probability Distributions  6.1.4. Expectation and Variance  6.1.5. Conditional Probability and Bayes' Theorem  6.2. Statistics  6.2.1. Descriptive Statistics  6.2.2. Inferential Statistics  6.2.3. Hypothesis Testing  6.2.4. Regression Analysis  6.2.5. ANOVA
  7. Numerical Methods  7.1. Root-Finding Algorithms  7.2. Numerical Integration  7.3. Numerical Differentiation  7.4. Numerical Solutions to Differential Equations  7.5. Matrix Computations  7.6. Interpolation and Extrapolation
  8. Discrete Mathematics  8.1. Logic and Proof  8.2. Combinatorics  8.3. Graph Theory  8.4. Boolean Algebra  8.5. Recurrence Relations
  9. Linear Programming and Optimization  9.1. Simplex Method  9.2. Duality  9.3. Integer Programming  9.4. Nonlinear Optimization
  10. Topology  10.1. Metric Spaces  10.2. Topological Spaces  10.3. Continuity and Homeomorphisms  10.4. Compactness  10.5. Connectedness
  11. Geometry  11.1. Euclidean Geometry  11.2. Differential Geometry  11.3. Non-Euclidean Geometry  11.4. Projective Geometry
  12. Mathematical Physics  12.1. Fourier Series  12.2. Laplace Transforms  12.3. Applications of Differential Equations  12.4. Special Functions
  13. Set Theory and Logic  13.1. Sets and Subsets  13.2. Relations and Functions  13.3. Cardinality  13.4. Formal Logic  13.5. Zermelo-Fraenkel Set Theory
  14. Functional Analysis  14.1. Normed Spaces  14.2. Banach Spaces  14.3. Hilbert Spaces  14.4. Linear Operators

Undergraduate → Algebra → Linear Algebra ↓


Systems of Linear Equations


Systems of linear equations are an essential part of linear algebra, a field of mathematics that deals systematically with equations and matrices. Linear equations themselves are differential equations in their first degree, expressed in terms of variables that do not involve any powers or products of these variables. Simply put, they are straight lines when graphed in two dimensions.

A system of linear equations is a set composed of two or more linear equations. The purpose of dealing with these systems is to find solutions, which means values for the unknown variables that satisfy all the equations simultaneously.

Understanding linear equations

To better understand systems of linear equations, it's important to first understand what a single linear equation looks like. Typically, a linear equation in two variables, such as x and y, looks something like this:

    ax + by = c

In this equation, a, b, and c are constants, and x and y are variables. The graph of this equation is a straight line in the xy-plane.

Example of a system of linear equations

A simple example of a system with two linear equations is:

    1. x + y = 5
    2. x – y = 1

The solutions of these two equations are the values of x and y that make both the equations true at the same time.

Graphical representation

One way to solve systems of equations is through a graphical representation. Here is a representation of the equations above:

    Equation 1: y = -x + 5
    Equation 2: y = x – 1
y = -x + 5 y = x – 1

In the graph above, the blue line represents the first equation, and the green line represents the second equation. The point at which they intersect is the solution to the system of equations. By visually examining the graph, we can determine that the lines intersect at the point (3, 2). Therefore, x = 3 and y = 2 are the solutions to the system.

Algebraic methods for solving systems

Since it may be difficult to find the exact intersection by simply graphing, algebraic methods such as substitution and elimination provide more accurate solutions.

Replacement method

The substitution method involves solving one of the equations for a variable and substituting that expression into the other equation. Let's apply it to our example:

  1. Solve the first equation for x: x = 5 - y
  2. Substitute this expression into the second equation: (5 - y) - y = 1
  3. Simplify and solve: 5 - 2y = 1
  4. -2y = 1 - 5 = -4, so y = 2
  5. Substitute y in the expression for x: x = 5 - 2 = 3

Elimination method

The elimination method involves adding or subtracting equations from each other in such a way that one of the variables is eliminated. In our example it works like this:

  1. Align the equations: 
                1. x + y = 5
            2. x – y = 1
  2. Add the two equations together to eliminate y: 
                x+y
          + x - y
          ,
            2x = 6
  3. Solve for x: x = 6 / 2 = 3
  4. Substitute x into one of the original equations to find y: 3 + y = 5, so y = 2

Types of solutions

Linear equation systems can have different types of solutions depending on the relationship between the lines.

1. Unique solution

This happens when the lines intersect at a single point. The coefficients in the equations provide different slopes:

    Example: 
    1. 2x + 3y = 6
    2. x – y = 2

These lines intersect each other exactly once, and leave a unique solution, such as the point (x, y)

2. Infinite solutions

When two equations represent the same line, the system has infinitely many solutions. In this case, every solution to one equation is a solution to the other:

    Example:
    1. 2x + 2y = 4
    2. x + y = 2

Both the equations represent the same line.

3. No solution

There is no solution when parallel lines are involved. Parallel lines will never meet, so there is no general solution:

    Example: 
    1. x – y = 1
    2. x – y = 3

These lines have the same slope but different y-intercepts.

Matrix solution approach

A more systematic approach to handling large systems of linear equations involves matrices. A system can be represented compactly using a matrix. Let's take an example to explore how matrices work with systems of equations:

    System:
    1.x + 2y = 5
    2.3x + 4y = 6
    
    Matrix representation:
    a * x = b
    
    A = [1 2]
        [3 4]

    x = [x]
        [y]

    B = [5]
        [6]

The solution is given by finding the inverse of the matrix A and solving for X:

    x = a - 1 * b

However, finding the inverse is only possible if the determinant of A is not zero.

This approach is especially effective for systems with more than two variables.

Conclusion

Solving systems of linear equations is a foundational skill in linear algebra that is crucial for countless real-world applications. Understanding the properties and different ways to solve these systems allows for creative and efficient problem-solving approaches. As you study deeper, practicing with larger systems and using matrix representations will enhance your skills and open up new avenues for applying these fundamental principles. Remember, the beauty of linear algebra lies in its simplicity and universality, providing elegant solutions to complex problems.


Undergraduate → 1.1.3


U
username
0%
completed in Undergraduate

← Prev (1.1.2)
Determinants
Next (1.1.4) →
Vector Spaces

Comments 



Systems of Linear Equations

https://www.buddymath.com/ug/1.1.3?bmal=en