$\begin{array}{l}
-2 x+y=3 \\
2 x-y=4
\end{array}$

Using elimination to solve this system results in the equation $\square$

Add the two equations together: ( − 2 x + y ) + ( 2 x − y ) = 3 + 4 .
Simplify the left side: − 2 x + 2 x + y − y = 0 .
Simplify the right side: 3 + 4 = 7 .
The resulting equation is 0 = 7 , indicating no solution exists: 0 = 7 ​ .

Explanation

Analyzing the System of Equations We are given the following system of equations:

− 2 x + y = 3 2 x − y = 4 ​
We want to use the elimination method to solve this system.

Eliminating Variables To use the elimination method, we can add the two equations together. Notice that the terms − 2 x and 2 x cancel each other out, and the terms y and − y also cancel each other out.

Combining the Equations Adding the left-hand sides of the two equations, we get:


( − 2 x + y ) + ( 2 x − y ) = − 2 x + y + 2 x − y = 0
Adding the right-hand sides of the two equations, we get:
3 + 4 = 7
Therefore, when we add the two equations together, we obtain the equation:
0 = 7
This equation indicates that the system of equations has no solution, since 0 cannot equal 7 .

Final Equation The equation resulting from the elimination method is 0 = 7 . This indicates that the system of equations is inconsistent and has no solution.

Examples
Consider a scenario where you're trying to balance a chemical equation. If, after combining equations, you arrive at an inconsistency like 0 = 7 , it means the original equations are contradictory, and there's no valid solution that satisfies all conditions. This principle applies to various fields, including circuit analysis, economics, and resource allocation, where inconsistent equations indicate an impossible or unsolvable situation.

Answered by GinnyAnswer | 2025-07-08