๐ What is a Consistent System of Linear Equations?
In the realm of linear algebra, a consistent system of linear equations refers to a set of two or more equations that possess at least one solution. In simpler terms, the lines, planes, or hyperplanes represented by these equations intersect at one or more points. If a solution exists, regardless of whether it's unique or infinite, the system is considered consistent.
๐ A Brief History
The study of linear equations dates back to ancient civilizations, with early methods for solving them appearing in Babylonian and Egyptian mathematics. The modern systematic approach to linear algebra emerged in the 18th and 19th centuries, with mathematicians like Carl Friedrich Gauss developing methods such as Gaussian elimination, which are instrumental in determining the consistency of a system of equations.
๐ Key Principles of Consistent Systems
- ๐ Graphical Representation: A consistent system of two linear equations in two variables will have lines that either intersect at a single point (unique solution) or coincide (infinite solutions).
- ๐งฎ Algebraic Condition: For a system to be consistent, the equations must not contradict each other. This can be checked using methods like substitution, elimination, or matrix operations.
- ๐งฉ Matrix Representation: In matrix form ($Ax = b$), a system is consistent if the rank of the coefficient matrix ($A$) is equal to the rank of the augmented matrix ($[A|b]$).
- โพ๏ธ Infinite Solutions: A consistent system has infinitely many solutions when the equations are dependent, meaning one equation can be obtained by multiplying the other by a constant.
- ๐ฏ Unique Solution: A consistent system has a unique solution when the lines intersect at only one point.
๐ Real-world Examples
- ๐ฐ Budget Allocation: Imagine allocating a budget between two departments. If there are constraints on total spending and minimum spending for each department, the equations representing these constraints must form a consistent system to have a feasible budget plan.
- ๐งช Chemical Reactions: Balancing chemical equations involves finding coefficients that satisfy the conservation of mass. A consistent system of equations ensures that a balanced reaction exists.
- ๐บ๏ธ Navigation: Determining the location of a ship using multiple radar stations involves solving a system of equations. The system must be consistent to accurately pinpoint the ship's position.
๐ Determining Consistency
Several methods can be used to determine if a system is consistent:
- โ Substitution: Solve one equation for one variable and substitute that expression into the other equation(s). If you arrive at a true statement (e.g., 0 = 0), the system is consistent and dependent. If you arrive at a contradiction (e.g., 0 = 1), the system is inconsistent.
- โ Elimination: Add or subtract multiples of the equations to eliminate one variable. If you can eliminate all variables and arrive at a true statement, the system is consistent and dependent. If you arrive at a contradiction, the system is inconsistent.
- ๐ Graphical Method: Graph the equations. If the lines intersect, the system is consistent. If they are parallel and distinct, the system is inconsistent. If they are the same line, the system is consistent and dependent.
- ๐ข Matrix Method: Write the system as $Ax = b$. The system is consistent if and only if $b$ is in the column space of $A$, which means the rank of $A$ is equal to the rank of $[A|b]$.
๐ข Example Problems and Solutions
Example 1: Consider the system:
$\begin{cases}
x + y = 5 \\
2x + 2y = 10
\end{cases}$
Dividing the second equation by 2, we get $x + y = 5$, which is identical to the first equation. Thus, the system is consistent and dependent (infinite solutions).
Example 2: Consider the system:
$\begin{cases}
x + y = 3 \\
x - y = 1
\end{cases}$
Adding the two equations gives $2x = 4$, so $x = 2$. Substituting into the first equation, $2 + y = 3$, so $y = 1$. Thus, the system has a unique solution $(2, 1)$ and is consistent.
Example 3: Consider the system:
$\begin{cases}
x + y = 1 \\
x + y = 5
\end{cases}$
Subtracting the first equation from the second gives $0 = 4$, which is a contradiction. Thus, the system is inconsistent.
๐ Advanced Concepts
- ๐งฎ Rank and Nullity: The consistency of a system is deeply connected to the rank of the coefficient matrix and the concept of nullity. Understanding these concepts provides deeper insights into the solvability and nature of solutions.
- ๐ Linear Independence: A consistent system implies that the equations are linearly independent or dependent. Linear independence is crucial in determining if the solution is unique.
๐ Conclusion
Understanding consistent systems of linear equations is fundamental to many areas of mathematics, science, and engineering. Whether you're solving a system by hand or using computational tools, recognizing the conditions for consistency is essential for obtaining meaningful solutions.