Equivalent Systems of Equations
An equivalent system is a set of linear equations that shares the same solution set as another set of linear equations.
In other words, two systems of equations are equivalent if they yield the same solutions when solved.
The elementary operations that can be used to transform one system of linear equations into an equivalent system, without changing the solution set, include:
- Adding to an equation another equation of the system multiplied by a number.
- Multiplying an equation by a non-zero number.
- Swapping the positions of two equations.
- Removing or adding an identity (an equation like 0 = 0).
These operations are fundamental for manipulating systems of linear equations because they allow simplification without losing information about the solutions, especially when using methods like Gaussian elimination.
Example
Consider the following system of linear equations:
\[ S: \begin{cases}
2x + 3y = 5 \\
4x + 6y = 10
\end{cases} \]
To determine if we can find an equivalent system, let's apply some elementary operations:
We can divide the second equation by 2:
$$ S: \begin{cases} 2x + 3y = 5 \\ \frac{4x + 6y}{2} = \frac{10}{2} \end{cases} $$
$$ S: \begin{cases} 2x + 3y = 5 \\ 2x + 3y = 5 \end{cases} $$
We notice that after the division, the second equation is identical to the first. Therefore, we can rewrite the system \( S \) as:
\[ S': \begin{cases} 2x + 3y = 5 \\ 2x + 3y = 5 \end{cases} \]
We can further simplify the system by eliminating the redundant equation (one of the identical equations):
\[ S'': \begin{cases}
2x + 3y = 5
\end{cases} \]
Explanation. If we add the first equation multiplied by -1 to the second equation, we get an identity, which can be removed from the system. \[ S'': \begin{cases} 2x + 3y = 5 \\ 2x + 3y - (2x+3y) = 5 - (5) \end{cases} \] \[ S'': \begin{cases} 2x + 3y = 5 \\ 0 = 0 \end{cases} \] \[ S'': \begin{cases} 2x + 3y = 5 \end{cases} \]
The system \( S'' \) is equivalent to the original system \( S \) because they have exactly the same solutions.
\[ S'': \begin{cases}
2x + 3y = 5
\end{cases} \]
Indeed, any pair \( (x, y) \) that satisfies the equation \( 2x + 3y = 5 \) will also satisfy the original system \( S \).
Thus, we have transformed the system \( S \) into an equivalent system \( S'' \) using elementary operations without altering the solution set of the system.
Properties of Equivalent Systems
The fundamental properties of equivalence between systems of equations are:
- Reflexive Property: every system is equivalent to itself.
- Symmetric Property: if one system is equivalent to another, then the second is also equivalent to the first.
- Transitive Property: if a system is equivalent to a second system and the second system is equivalent to a third, then the first system is equivalent to the third.
These properties confirm that equivalence between systems of equations is an equivalence relation.
