Verified Answer :

  • Subtract equations to eliminate \(z\), reducing the problem to a system of two equations: \(x + y = -1\) and \(5x + 2y = 7\).
  • Solve the resulting system for \(x\) and \(y\) to find \(x=3\) and \(y=-4\).
  • Substitute these values back into one of the original equations to determine \(z=-10\).
  • The solution is \(x = \boxed{3}, y = \boxed{-4}, z = \boxed{-10}\).

Explanation

  1. Problem Analysis

We are given a system of three linear equations with three variables \(x\), \(y\), and \(z\):
1) \(4x + 3y - z = 10\)
2) \(8x + 7y - z = 6\)
3) \(6x + y + z = 4\)

Our goal is to determine the unique values of \(x\), \(y\), and \(z\) that satisfy all three equations simultaneously. We will use the method of elimination to solve this system.

  1. Eliminating the Variable z

To eliminate the variable \(z\), we can first subtract equation (1) from equation (2):
\((8x + 7y - z) - (4x + 3y - z) = 6 - 10\)
\(4x + 4y = -4\)
Dividing by 4, we get a simplified linear equation:
(4) \(x + y = -1\)

Next, we eliminate \(z\) by adding equation (1) and equation (3):
\((4x + 3y - z) + (6x + y + z) = 10 + 4\)
\(10x + 4y = 14\)
Dividing by 2, we get a second simplified linear equation:
(5) \(5x + 2y = 7\)

  1. Solving for x, y, and z

Now we have a smaller system of two linear equations with two variables:
(4) \(x + y = -1\)
(5) \(5x + 2y = 7\)

From (4), we can write \(y = -1 - x\). Substituting this into equation (5):
\(5x + 2(-1 - x) = 7\)
\(5x - 2 - 2x = 7\)
\(3x - 2 = 7\)
\(3x = 9\)
\(x = 3\)

Now, substitute \(x = 3\) back into the expression for \(y\):
\(y = -1 - (3) = -4\)

Finally, substitute \(x = 3\) and \(y = -4\) into original equation (3) to find \(z\):
\(6(3) + (-4) + z = 4\)
\(18 - 4 + z = 4\)
\(14 + z = 4\)
\(z = -10\)

  1. Verification and Final Result

We have found the values \(x = 3\), \(y = -4\), and \(z = -10\). Checking these values in the original equations:
1) \(4(3) + 3(-4) - (-10) = 12 - 12 + 10 = 10\) (Correct)
2) \(8(3) + 7(-4) - (-10) = 24 - 28 + 10 = 6\) (Correct)
3) \(6(3) + (-4) + (-10) = 18 - 4 - 10 = 4\) (Correct)

Thus, the solution is \(x = 3\), \(y = -4\), \(z = -10\).

Examples

This method of solving systems of linear equations is crucial in fields like civil engineering. For example, when calculating the forces acting on a truss bridge, engineers set up equations based on equilibrium principles (where the sum of forces and moments must equal zero). Solving these systems allows them to determine the tension or compression in each individual beam, ensuring the structural integrity and safety of the bridge for daily use.

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