# Systems of Equations

Systems of equations are sets of equations where the solution is the intersecting point or points between the equations. Most of the systems of equations seen in algebra are sets of two linear equations in the standard form i.e Ax + By = C, here A, B, and C are constant numbers and x and y are variables.

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## Solutions to System of Equations

Solving a system of equations signifies finding the values of the variables used in the set of equations. We calculate the values of the unknown variables while balancing the equations on both sides. The main reason behind solving an equation system is to find the value of the variable that meets the condition of all the given equations true. There can be various types of solutions to a given system of equations. Following are the three major ones.

- Unique solution
- No solution
- Infinitely many solutions

## Methods to Solve System of Equations

There are generally three methods used to solve systems of linear equations. All three methods can be used to solve more complex linear equations as well.

Following are the three:

- The Graphing Method,
- The Substitution Method, and
- The Elimination Method.

Note: To solve a system of equations in 2 variables, we need at least 2 equations. Similarly, for solving a system of equations in ānā variables, we will require at least ānā equations.

## Graphing Method

In this method, equations are solved by graphing each equation in the system and specifying the point(s) of the intersection. It is comparatively easier to graph the equations by transforming them to slope-intercept form.

## Substitution Method

In this method, equations are solved by replacing a variable in one equation with the equal of that variable, calculated using the other equation.

The first step of this method is to solve for a variable in one equation. By examining each coefficient we can decide which variable to solve. Mostly variables with coefficients of 1 are the easiest to solve as well need not divide by anything.

## Elimination Method

In this method, equations are solved by eliminating one variable. Before starting with the elimination, it is essential that both equations are in the standard form Ax + By = C. This classifies the systems of equations by aligning each term in one equation with their corresponding term in the other.

## Example

### Substitution Method

Here, the substitution method gives the result as a true statement, 9 = 9, the system of equations has infinite solutions.

### Elimination Method

Like the substitution method, here the elimination method gives another true statement, 0 = 0, the system of equations has infinite solutions.

### Graphing Method

The graph of the two equations overlaps each other, this is because the system of equations defines the same line, guaranteeing that there are infinite solutions to both equations.

## Applications of System of Equations

Applications of linear equations can be seen in our surroundings and can be used by people on a daily basis because the situations faced by them might have an unknown quantity that can be defined as a linear equation such as calculating mileage rates, income over time, etc.

The main goal for the applications of linear equations or linear systems is to solve diverse problems using two variables where one is known and the other is unknown, likewise dependent on the first.

Following are some of the applications of linear equations:

- Application of linear equation in industry and economics
- Finances problems by using two variables
- Geometry problems by using two variables.
- Distance-Rate-Time problems by using two variables.