## System of Linear Equations theory

# System of Linear Equations Theory: First Degree

The simplest method for solving a System of Linear Equations is to repeatedly eliminate variables.

Steps to take:

1. In the first equation, solve the equation for a variable relative to the rest of the equation

- Enter this expression in the other equations. This will leave a system of equations with an equation with a less variable.

- Repeat this method until a single linear equation with a single variable

- Solve this equation

- Replaces the result by the variable in the above equations until the final equation is resolved

# System of Equations Theory: Second Degree

The simplest method for solving a System equations is to repeatedly eliminate variables.

Steps to take:

1. In the first equation, solve the equation for a variable relative to the rest of the equation

- Enter this expression in the other equations. This will leave a system of equations with an equation with a less variable.

- Repeat this method until a single linear equation with a single variable

- Solve applying the next formula:

# System Linear Equations with Two Unknowns Theory

There are several different methods which will lead to obtaining the same results:

- Substitution method.
- Equalization method.
- Reduction method.
- Gauss method.
- The inverse matrix method.
- Cramer’s rule.

__Substitution method:__

- Solve one of the unknowns in one of the equations.
- The expression of this mystery is replaced in the other equation, thereby obtaining an equation with a single unknown.
- The equation obtained is solved.
- The value obtained is substituted in the equation that appeared clear-unknown.
- This equation is solved and the two values obtained are the system solution.

__ Example:__

We solve for the x unknown in the second equation:

And it is replaced in the first equation:

It has already solved the linear equation with one unknown:

The value obtained is substituted into the first equation we have solved:

**SOLUTION:**

x= 1; y= 3

__Equalization method:__

1. Solve one of the unknowns in both equations.

2. The expressions obtained are equaled and we obtain an equation with one unknown

3. The equation is solved.

4. The value obtained is substituted in either of the two expressions that appeared clear the other unknown.

5. The two values obtained are the system solution.

__Example: __

We solve the same unknown in both equations:

We equate the two expressions:

Substituting it in either expression cleared first:

**SOLUTION:**

X= 4; y= -3

__Reduction method:__

- Equations are prepared, multiplied them by the appropriate number.

2. Subtract them to simplify and thus one of the unknowns disappears.

3. The resulting equation is solved.

4. The value obtained is substituted in one of the initial equations and it is solved.

5. The two values obtained are the system solution.

__Example:__

The equations are prepared to be subtracted. The unknown “y” is suppressed, staying:

The value obtained in one of the initial equations is replaced:

**SOLUTION:**

x= 2; y=3

**This theory about System of Linear Equations continue on the next post “System of Equations theory II”**

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