Linear Equations and Systems of Equations

INTRODUCTION

Linear equations and systems of equations are basic mathematical expressions where the unknowns appear in the first degree (i.e., raised to the power of 1). Linear equations and systems of equations allow for the determination of the values of variables and are key to understanding algebraic relationships between quantities.

LINEAR EQUATIONS

A linear equation is an equation of the form ax + b = c, where a, b, and c are real numbers, and x is the unknown. The goal is to determine the value of x that satisfies the equation.

Example of solving:
Given: 2x + 5 = 11
Subtract 5 from both sides: 2x = 11 - 5
2x = 6
Divide by 2: x = 3

Linear equations can have one solution, infinitely many solutions, or no solution, depending on the coefficients.

SYSTEMS OF LINEAR EQUATIONS

A system of linear equations is a group of two or more linear equations that share common unknowns. The form of a system with two unknowns is:

a₁x + b₁y = c₁
a₂x + b₂y = c₂

Systems of equations are solved using three main methods:

1. SUBSTITUTION METHOD: Express one unknown from one equation and substitute it into the other equation.
2. ELIMINATION METHOD (BY EQUATING COEFFICIENTS/EQUALIZATION): Transform both equations so that one variable has the same coefficient (or opposite coefficients) in both, allowing it to be eliminated by subtraction or addition. The Slovenian text mentions "Metoda izenačevanja" which often implies making one variable the subject of both equations and then equating those expressions. This also falls under the broader category of elimination or comparison.
3. ELIMINATION METHOD (BY ADDITION/SUBTRACTION): Add or subtract the equations (possibly after multiplying them by constants) to eliminate one unknown.

IMPORTANCE OF LINEAR EQUATIONS AND SYSTEMS

Linear equations and systems of equations allow for the analysis of relationships between quantities and are the basis for many mathematical and practical problems. Solving them is crucial for further studies in algebra.

CONCLUSION

Linear equations and systems of equations are fundamental mathematical expressions for determining the values of unknowns. Understanding them enables the effective solving of numerical and algebraic problems.