System of Two Linear Equations 1

A system of two linear equations in mathematics is a collection of two equations with two unknowns that must be solved together to find their common solutions. These equations are typically presented in the basic form ax + by = c and dx + ey = f, where a, b, c, d, e, and f are constants, and x and y are the unknowns we are looking for.

WHY DO WE DEAL WITH SYSTEMS OF LINEAR EQUATIONS?

Systems of linear equations are fundamental in algebra and have numerous applications in the real world, such as solving problems in physics, economics, and engineering. They allow us to find the values of unknowns that simultaneously satisfy more than one linear equation.

METHODS OF SOLVING

Various methods exist for solving a system of two linear equations:

1. GRAPHICAL METHOD: This involves drawing the graphs of both linear equations on a coordinate system. The point where the graphs intersect represents the solution to the system.
2. SUBSTITUTION METHOD: This requires solving one equation for one unknown and then substituting that value into the other equation, resulting in a single equation with one unknown.
3. ELIMINATION METHOD (OR ADDITION/SUBTRACTION METHOD): This involves adding or subtracting the equations (possibly after multiplying them by constants) to eliminate one of the unknowns, allowing us to solve the system for the remaining unknown.

MEANING OF THE SOLUTION

The solution to a system of two linear equations is a pair of values, x and y, which, when substituted into both equations, make both equations true. A system can have:

• ONE SOLUTION: The point where the graphs intersect.
• NO SOLUTION: If the graphs are parallel and never intersect.
• INFINITELY MANY SOLUTIONS: If the graphs are identical (the lines coincide).

CONCLUSION

A system of two linear equations is a basic concept in mathematics that enables students to understand how to solve problems requiring the simultaneous satisfaction of more than one constraint or condition. By understanding and using various methods for solving these systems, students learn logical reasoning and analytical skills that are useful in many fields. Understanding these concepts is key for anyone wanting to develop a solid foundation in algebra and continue with the study of mathematics.