Equivalent Systems of Equations
Worked Example
Problem: Show that System A and System B are equivalent systems of equations.
System A: x + y = 5 and x − y = 1
System B: 2x = 6 and x − y = 1
Step 1: Solve System A. Add the two equations to eliminate y.
(x+y)+(x−y)=5+1⟹2x=6⟹x=3
Step 2: Substitute x = 3 back into x + y = 5 to find y.
3+y=5⟹y=2
Step 3: Solve System B. From 2x = 6, get x = 3. Substitute into x − y = 1.
3−y=1⟹y=2
Step 4: Compare the solution sets. Both systems have the unique solution (3, 2).
Answer: Both systems have the solution (x, y) = (3, 2), so they are equivalent systems of equations.
Why It Matters
When you solve a system of equations, each algebraic step—such as adding two equations together or multiplying an equation by a nonzero constant—produces a new system equivalent to the original. Recognizing this guarantees that the solution you find at the end actually satisfies the equations you started with. This concept underpins methods like Gaussian elimination and substitution.
Common Mistakes
Mistake: Assuming two systems are equivalent just because they have one solution in common.
Correction: Equivalent systems must share every solution. For instance, a system with infinitely many solutions is not equivalent to one with a single solution, even if that single solution happens to satisfy both systems.
Related Terms
- Simultaneous Equations — Equations solved together as a system
- Solution — Values that satisfy every equation in a system
- System of Equations — A set of equations considered together
- Gaussian Elimination — Method that produces equivalent systems step by step
- Equivalent Equations — Single equations sharing the same solution set