Lab Report: Solving Systems of Linear Equations

Introduction

In mathematics, a system of linear equations (or linear system) is a collection of one or more linear equations involving the same set of variables.

Math Introduction:

A system of linear equations can be represented as follows:

$$


3x
+
2y
-
z
=
1


2x
-
2y
+
4z
=
-2


-x
+
1/2 y
-
z
=
0

Linear Algebraic Equation:

An equation of the form F(x₁, x₂, …, x_n) = 0 is a general set of equations with n equations and n unknowns.

Matrix Form: [A] {x} = {b}

Where:

• [A] is an n x n coefficient matrix
• {x} is an n x 1 unknown vector
• {b} is an n x 1 right-hand-side (RHS) vector

Review of Matrices:

[A] =

Square matrix: [A] nxm is a square matrix if n=m.

Main (principle) diagonal: The main diagonal of [A] nxn consists of elements a_ii; i=1, ..., n.

Symmetric matrix: If a_ij = a_ji, [A]nxn is a symmetric matrix.

Diagonal matrix: [A]nxn is diagonal if a_ij = 0 for all i=1, ..., n ; j=1, ..., n and i≠j.

Identity matrix: [A]nxn is an identity matrix if it is diagonal with a_ii=1 for i=1... n. Shown as [I].

In mathematics, the hypothesis of a linear system is the basic and principal part of direct variable-based math, a subject used in many parts of modern mathematics. Computational algorithms for finding the solutions are a significant part of numerical linear algebra and play a prominent role in engineering, physics, chemistry, computer science, and economics. A system of nonlinear equations can often be approximated by a linear system, a useful technique when creating a mathematical model or computer simulation of a relatively complex system.

Oftentimes, the coefficients of the equations are real or complex numbers, and the solutions are sought in the same set of numbers. However, the theory and the algorithms apply to coefficients and solutions in any field. For solutions in an integral domain like the ring of integers or in other algebraic structures, various theories have been developed; see linear equation over a ring. Integer linear programming is a collection of methods for finding the 'best' integer solution (when there are many).

Gröbner basis theory provides algorithms when coefficients and variables are polynomials.

Additionally, tropical geometry is an example of linear algebra in a more exotic form.

Discussion

A system of linear equations is a set of multiple linear equations. The goal is to find the ordered n-tuple that satisfies all n equations. Therefore, that ordered n-tuple is the solution of that system. For example, an ordered pair (x, y) that satisfies both equations of the system is a solution of that system.

There are four methods for solving systems of equations:

1. Substitution Method:

The Substitution Method is a straightforward technique where one equation is solved for one variable, and then that variable is substituted into the other equation and solved. Let's learn the Substitution method for finding a solution to a system of linear equations in two variables.

Example 1:

Solve the following system of linear equations using the substitution method:

Equation A: 2x + 3y = 7

Equation B: x - 2y = 4

Substitute x - 2y from Equation B into Equation A:

2(x - 2y) + 3y = 7

Simplify and solve the equation for x:

2x - 4y + 3y = 7

2x - y = 7

Solve for y:

-y = 7 - 2x

y = 2x - 7

You can substitute a value for a variable even if it is an expression.

Example 2:

Solve for x and y:

Equation A: 3x + 2y = 10

Equation B: 2x - y = 3

The goal of the substitution method is to rewrite one of the equations in terms of a single variable. Equation B allows us to do that, so we substitute 2x - y into Equation A for x:

3(2x - y) + 2y = 10

Simplify and solve the equation for y:

6x - 3y + 2y = 10

6x - y = 10

Now, find x by substituting this value for y into another equation and solve for x using Equation B:

2x - (2x - 7) = 3

2x - 2x + 7 = 3

7 = 3

This is a false statement, which means there is no solution.

2. Identify Systems with No Solution or Infinite Solutions:

Just like when solving linear equations in one variable, some systems of equations have no solutions, while others have an infinite number of solutions. Let's explore these scenarios:

Example 3:

Solve for x and y:

Equation A: 2x + 3y = 6

Equation B: 4x + 6y = 12

First equation is simply a multiple of Equation A, so we can see that there are infinitely many solutions.

Example 4:

Solve for x and y:

Equation A: 3x + 4y = 7

Equation B: 6x + 8y = 14

These equations represent the same line when graphed, which means there are infinitely many solutions.

3. Solve a System of Equations Using the Elimination Method:

The elimination method for solving systems of linear equations uses the addition property of equality. You can add the same value to both sides of an equation to eliminate one of the variable terms. In this method, you may or may not need to multiply the terms in one equation by a number first. Let's first look at examples where no multiplication is necessary to use the elimination method.

Example 5:

If you add both equations together:

(2x + 3y) + (3x - 2y) = (5x + y) = 10

You have eliminated the y term, and this equation can be solved using the methods for solving equations with one variable.

Example 6:

Use elimination to solve the system:

Equation A: 2x - 3y = 5

Equation B: 4x + 6y = 10

Add the equations:

(2x - 3y) + (4x + 6y) = (6x + 3y) = 15

Now, substitute (6x + 3y) = 15 into one of the original equations and solve for y:

2x - 3y = 5

2x - 3(5) = 5

2x - 15 = 5

2x = 5 + 15

2x = 20

x = 20 / 2

x = 10

Be sure to check your answer in both equations!

Now, let's look at the addition method:

Example 7:

Solve the following system of linear equations using the addition method:

Equation A: 2x + 3y = 12

Equation B: 4x - 2y = 10

If we add both equations:

(2x + 3y) + (4x - 2y) = (6x + y) = 22

Now, we can solve for x:

6x + y = 22

6x = 22 - y

x = (22 - y) / 6

These are some methods for solving systems of linear equations, and they can yield a single solution, no solution, or infinitely many solutions, depending on the nature of the equations.

Conclusion

The substitution method is one way of solving systems of equations. To use the substitution method, one equation is solved for one variable in terms of the other variable. Then, that expression is substituted into the second equation. Solving using the substitution method will yield one of three results: a single value for each variable within the system (indicating one solution), a false expression (indicating no solutions), or a true statement (indicating an infinite number of solutions).

Combining equations is a powerful tool for solving a system of equations. Adding or subtracting two equations to eliminate a common variable is known as the elimination (or addition) method. Once one variable is eliminated, it becomes much easier to solve for the other one. Multiplication can be used to set up matching terms in equations before they are combined to help find a solution to a system.

When using the elimination method, it is important to multiply all the terms on both sides of the equation—not just the one term you are trying to eliminate.