Verified Answer :

  • Identify the system of equations: \(2x + y = 4\) and \(-2x + y = -4\).
  • Add the equations to eliminate \(x\), which gives \(2y = 0\), so \(y = 0\).
  • Substitute \(y = 0\) into the first equation: \(2x + 0 = 4\), which gives \(x = 2\).
  • The solution to the system is \(\boxed{x=2, y=0}\).

Explanation

  1. Analysis of the system of equations

The problem asks us to solve a system of two linear equations in two variables, \(x\) and \(y\). The given equations are:
1) \(2x + y = 4\)
2) \(-2x + y = -4\)
We observe that this is a system of linear equations, and we can use the method of elimination because the coefficients of \(x\) are \(2\) and \(-2\), which are additive inverses. Adding these equations will eliminate \(x\).

  1. Solving for y by elimination

To eliminate \(x\), add equation (1) and equation (2):
\((2x + y) + (-2x + y) = 4 + (-4)\)
Combine the terms:
\((2x - 2x) + (y + y) = 0\)
\(2y = 0\)
Divide by \(2\):
\(y = 0\)

  1. Solving for x by substitution

Now substitute \(y = 0\) into equation (1) to solve for \(x\):
\(2x + (0) = 4\)
\(2x = 4\)
Divide by \(2\):
\(x = 2\)

  1. Verification and conclusion

Finally, verify the solution \((x, y) = (2, 0)\) by substituting into equation (2):
\(-2(2) + (0) = -4 + 0 = -4\)
This matches equation (2), so the solution is correct. The final answer is \(x = 2\) and \(y = 0\).

Examples

Systems of equations like this are used in many real-life situations, such as finding the equilibrium point where the supply and demand curves intersect. If we represent demand as \(y = -2x + 4\) and supply as \(y = 2x - 4\), solving for \(x\) and \(y\) tells us the price (\(x\)) and quantity (\(y\)) where the market is balanced. Understanding this helps businesses make better decisions about pricing and production levels.

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