Solution of a Linear System

The solution to a linear system is a set of values for the unknowns that satisfies all equations in the system simultaneously.

Highlight: A solution to a linear system must:

1. Be an ordered pair (for two unknowns)
2. Consist of real numbers
3. Satisfy all equations in the system

Example: For a system with two unknowns, a solution might be represented as (7, 3), where x = 7 and y = 3.

Types of Linear Systems

Linear systems can be classified into three main categories based on their solutions:

1. Sistema determinato (Determined system): Has a finite number of solutions.
2. Sistema indeterminato (Undetermined system): Has an infinite number of solutions.
3. Sistema impossibile (Impossible system): Has no solution.

Highlight: Understanding the type of system is crucial for choosing the appropriate solving method and interpreting the results.

Methods for Solving Linear Systems

There are several methods to solve linear systems. This guide covers four main approaches:

1. Sistemi lineari - metodo di sostituzione (Substitution method)
2. Sistemi lineari metodo di confronto (Comparison method)
3. Sistemi lineari metodo di riduzione (Reduction method)
4. Cramer's method

Each method has its advantages and is suitable for different types of systems.

Substitution Method

The metodo di sostituzione is a straightforward approach to solving linear systems.

Steps:

1. Express one variable in terms of the other from one equation.
2. Substitute this expression into the other equation.
3. Solve for the remaining variable.
4. Substitute back to find the value of the first variable.

Example: For the system {x + y = 10, x - y = 4}:

1. From the first equation: x = 10 - y
2. Substitute into the second equation: $10 - y$ - y = 4
3. Solve: 10 - 2y = 4, y = 3
4. Substitute back: x = 10 - 3 = 7

The solution is (7, 3).

Comparison Method

The metodo di confronto involves comparing expressions for the same variable derived from different equations.

Steps:

1. Express one variable in terms of the other from both equations.
2. Set these expressions equal to each other.
3. Solve the resulting equation for one variable.
4. Substitute to find the other variable.

Example: For the system {2x + y = 1, x + y = 4}:

1. From both equations: y = 1 - 2x and y = 4 - x
2. Set equal: 1 - 2x = 4 - x
3. Solve: -x = 3, x = -3
4. Substitute: y = 4 - (-3) = 7

The solution is (-3, 7).

Reduction Method

The metodo di riduzione aims to eliminate one variable by adding or subtracting equations.

Steps:

1. Multiply equations to make coefficients of one variable equal in magnitude but opposite in sign.
2. Add or subtract the equations to eliminate that variable.
3. Solve for the remaining variable.
4. Substitute to find the eliminated variable.

Example: For the system {2x - 3y = 5, 5x + 3y = 9}:

1. Multiply the first equation by 5 and the second by 2: 10x - 15y = 25 10x + 6y = 18
2. Subtract: -21y = 7
3. Solve: y = -1/3
4. Substitute into 2x - 3y = 5 to find x = 2

The solution is (2, -1/3).

Cramer's Method

Cramer's method uses determinants to solve linear systems.

Steps:

1. Calculate the system determinant D.
2. Calculate determinants Dx and Dy by replacing columns with constants.
3. If D ≠ 0, the solution is x = Dx/D, y = Dy/D.

Highlight: Cramer's method also helps determine the system type:

• If D ≠ 0, the system is determined.
• If D = 0 and Dx or Dy ≠ 0, the system is impossible.
• If D = Dx = Dy = 0, the system is undetermined.

Example: For the system {2x - 5y = 4, 3x - 4y = 6}: D = 2(-4) - (-5)(3) = 7 Dx = 4(-4) - 6(-5) = 14 Dy = 2(6) - 4(4) = 0 x = 14/7 = 2, y = 0/7 = 0

The solution is (2, 0).

Special Cases and Exceptions

When dealing with linear systems, there are some special cases to consider:

1. Sistemi letterali (Literal systems): These systems contain variables as coefficients. They should be solved using Cramer's method, followed by a discussion of possible cases.

2. Sistemi fratti (Fractional systems): These systems involve fractions. It's important to find common denominators before solving.

Highlight: Always check for domain restrictions when dealing with fractional systems.

Criterion of Ratios

For systems in the form: {ax + by = c {a'x + b'y = c'

The criterion of ratios helps determine the system type without solving:

1. If a/a' = b/b' ≠ c/c', the system is determined.
2. If a/a' = b/b' = c/c', the system is undetermined.
3. If a/a' = b/b' and b/b' ≠ c/c', the system is impossible.

Highlight: This criterion provides a quick way to analyze the nature of a linear system before attempting to solve it.

Criterio dei Rapporti

Questa pagina finale introduce il criterio dei rapporti per analizzare i sistemi lineari. Si spiegano le condizioni per determinare se un sistema è determinato, impossibile o indeterminato basandosi sui rapporti tra i coefficienti delle equazioni.

Definizione: Il criterio dei rapporti è un metodo per determinare la natura di un sistema lineare senza risolverlo completamente. Highlight: Questo criterio è particolarmente utile per analizzare rapidamente la natura di un sistema senza dover eseguire calcoli complessi.