Verified Answer :

  • Identify the system of equations: \(5x + 7y = 11\) and \(5x - 2y = 29\).
  • Eliminate \(x\) by subtracting the second equation from the first, yielding \(9y = -18\), so \(y = -2\).
  • Substitute \(y = -2\) back into the first equation: \(5x - 14 = 11\), which simplifies to \(5x = 25\), giving \(x = 5\).
  • Final solution: \(x = \boxed{5}, y = \boxed{-2}\).

Explanation

  1. Analyze the System of Equations

We are given a system of two linear equations with two variables, \(x\) and \(y\):
1) \(5x + 7y = 11\)
2) \(5x - 2y = 29\)
To solve this, we can use the elimination method, which is efficient here because both equations contain the same \(5x\) term. By subtracting one equation from the other, we can eliminate \(x\) and solve for \(y\).

  1. Eliminate \(x\) and Solve for \(y\)

To eliminate \(x\), subtract Equation 2 from Equation 1:
\[(5x + 7y) - (5x - 2y) = 11 - 29\]
This simplifies to:
\[5x - 5x + 7y + 2y = -18\]
\[9y = -18\]
Dividing both sides by \(9\), we find:
\[y = \frac{-18}{9} = -2\]

  1. Solve for \(x\)

Now that we have the value of \(y\), we can substitute it into one of the original equations to solve for \(x\). Substituting \(y = -2\) into the first equation:
\[5x + 7(-2) = 11\]
\[5x - 14 = 11\]
Adding \(14\) to both sides, we get:
\[5x = 25\]
Dividing by \(5\), we find:
\[x = \frac{25}{5} = 5\]
Thus, the solution to the system is \(x = 5\) and \(y = -2\).

Examples

Systems of equations are used in various fields like economics to find the equilibrium point where supply equals demand. For instance, if you have a supply function \(S = 5x - 2\) and a demand function \(D = -7x + 11\), setting them equal allows you to determine the price \(x\) at which the market stabilizes. Similarly, this method helps in balancing chemical equations or planning budgets where multiple constraints must be met simultaneously. Mastering this technique provides a solid foundation for solving real-world optimization problems.

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