Solving Systems of Linear Equations with Elimination Method

System of Equations

A system of equations is a collection of two or more algebraic expressions with the same variables, such that their solutions satisfy all the given expressions simultaneously. These systems can be used in various fields, including physics, engineering, and economics, where they represent sets of equations that must hold true under specific conditions. In this article, we will discuss the elimination method, which helps solve systems of linear equations using different strategies.

The basic idea behind solving a system of linear equations is to manipulate the given equations so that one variable becomes isolated and its value can be determined by substituting into another equation. This process continues until all unknowns have been eliminated.

Elimination Method

One common technique to find the solution of a system of linear equations is the elimination method. This technique involves adding or subtracting coefficients of variables in order to eliminate them from the second set of equations, making it easier to solve for one variable. Once one variable is solved for, it can be substituted back into the original set of equations to solve for other variables.

To apply the elimination method, first choose any two equations containing the same variables. If necessary, multiply one or both equations by constants so that the coefficients of the eliminated variable become equal but opposite. Then, subtract the equation with higher degree from the equation with lower degree. This eliminates one variable, leaving you with only one variable in terms of that particular variable.

Example

Consider the following set of equations:

2x + y = 8
3x - y = 9

Rearranging the second equation by adding y to both sides gives:

3x - y + y = 9 + y

Next, adding the first equation to the rearranged second equation results in:

(2x + y) + (3x - y + y) = 8 + (9 + y)

This simplifies to:

5x = 18

Dividing both sides by 5, we get:

x = 3.6

Now, substitute x = 3.6 into either the first or second equation, say the second equation:

3(3.6) - y = 9 + y

Solving for y, we get:

y = 7.2

Thus, the solution is:

x = 3.6
y = 7.2

Linear Equations

Linear equations are equations in which each term has an exponent of zero or one. They typically take the form ax + b = c, where a, b, and c are constants and x represents the independent variable. Solving a single linear equation simply involves isolating the variable by performing operations like addition, subtraction, multiplication, or division that do not change the distribution of powers of the variable.

For example, consider the equation:

3x - 2 = 7

Subtracting 2 from both sides gives:

3x = 9

Then, dividing both sides by 3 gives:

x = 3

So, the solution is:

x = 3

In the context of systems of equations, a system of linear equations consists of multiple linear equations with the same variables. Each individual linear equation is called a main diagonal because when written in matrix form, it appears along the main diagonal line.

Elimination Strategy

When applying the elimination method to solve systems of linear equations, there are several strategies that can be employed depending on the nature of the coefficients and the equations involved:

1. Gaussian elimination: Also known as row reduction, Gaussian elimination is a systematic way of transforming a matrix into upper triangular or lower triangular form by elementary row operations.

2. Gauß-Jordan elimination: This is a variation of Gaussian elimination that transforms a matrix into identity form by adding or subtracting multiples of one row to another row.

3. Elimination by substitution: In this strategy, one equation is used to eliminate a variable from the other equation(s) by multiplying both sides of the first equation by a constant, and then adding or subtracting a multiple of the second equation to the first equation.

4. Elimination by elimination: Similar to elimination by substitution, elimination by elimination involves multiplying one equation by a constant and adding or subtracting a multiple of another equation to eliminate a variable.

By choosing the appropriate elimination strategy based on the given equations, we can efficiently solve the system of linear equations.