This document discusses systems of linear equations and methods for solving them. It defines a system of linear equations as two equations with two unknowns where the solution is a pair of numbers that satisfies both equations. Three methods for solving systems are presented: substitution, where one equation is used to solve for one unknown and substitutes it into the other; matching, where one unknown is solved for in both equations and set equal; and elimination, where the equations are manipulated to eliminate one unknown. The document also defines the three possible types of solutions to a system: a consistent independent system with a single solution, a consistent dependent system with infinite solutions, and an inconsistent system with no solution.

1. BILINGUAL SECTION – MARÍA ESTHER DE LA ROSA
EQUATIONS AND SYSTEM OF LINEAR EQUATIONS
1. EQUATIONS.
This is explained in the theory of the past year.
2. SYSTEM OF LINEAR EQUATIONS
Two equations with two unknowns form a system. The solution of a system is a pair of numbers such that
replacing x and y with its results both equations are verified.
Example: x = 2, y = 3
3. HOW TO SOLVE SYSTEMS OF LINEAR EQUATIONS
In this lesson you will learn how to solve systems of equations. You can use one of the following
methods.
• Substitution Method
• Matching Method
• Elimination Method
3.1 SUBSTITUTION METHOD
1. Work out the value of an unknown in one of the equations.
2. Substitute the expression of this unknown in the other equation, obtaining an equation with only one
unknown.
3. Solve the equation.
4. The value obtained is substituted into the other equation.
5. The two values obtained are the solution of the system.
Example 1: 2x – y = -9
x + y = 6
1Work out the value of x. x = 6 – y
2 Substitute into the first equation the value of x: 2(6 - y) – y = - 9
3 Solve the equation you have obtained: 12 – 2y –y = - 9; 12 – 3y = - 9; -3y = - 21; y = 7
4. To find x, return to x = 6 – y and replace y with 7
x = 6 – 7 = - 1

2. 3.2 MATCHING METHOD
1. Work out the value of an unknown in both equations.
2. Match the previous results, obtaining an equation with only one unknown.
3. Solve the equation.
4. The value obtained is substituted into one of the expression you have in the first pass.
5. The two values obtained are the solution of the system.
Example 2: 3x – 4y = -6
2x + 4y = 16
1 Work out the value of x.
2 Math both expressions:
3 Solve the equation you have obtained:
4. To find y, return to one of the iniitial expresions you have:
solution:
3.2 ELIMINATION METHOD
1. Prepare the two equations and multiply by the appropriate numbers in order to eliminate one of the
unknown values.
2. Add the systems and eliminate one of the unknowns.
3. Solve the resulting equation.
4, Substitute the value obtained into one of the initial equations and then solve.
5. The two values obtained are the solution of the system.
Example 3:
Replace the value of x in the second equation.

3. 4. TYPES OF SOLUTIONS SOLVING SYSTEMS OF LINEAR EQUATIONS
For two-variable systems, there are then three possible types of solutions:
A. Consistent Independent System
Always have a single solution.
x = 2, y = 3
B. Consistent Dependent System
The system has infinite solutions.
C. Inconsistent System
Has no solution