Verified Answer :
- Extract the three linear equations from the augmented matrix.
- Express one variable in terms of others using the simplest equation.
- Substitute the expression into the remaining equations to create a reduced system of two variables.
- Solve the reduced system for the variables and verify the results in the original equations to conclude \(\boxed{{(4, 1, -2)}}\).
Explanation
- Analyzing the System of Equations
The provided augmented matrix represents a system of three linear equations with three variables \(x, y, z\). The matrix corresponds to the following equations:
1) \(0x + 1y + 2z = -3\)
2) \(1x + 2y - 1z = 8\)
3) \(1x - 2y + 2z = -2\)
Our objective is to determine the unique solution \((x, y, z)\) that satisfies all three equations simultaneously.
- Reducing the System
We begin with the first equation, as it is the simplest to manipulate.
From equation (1):
\[y + 2z = -3\]
We can express \(y\) in terms of \(z\):
\[y = -3 - 2z\]
Next, we substitute this expression for \(y\) into equations (2) and (3) to reduce the system to two equations with two variables, \(x\) and \(z\).
For equation (2):
\[x + 2(-3 - 2z) - z = 8\]
\[x - 6 - 4z - z = 8\]
\[x - 5z = 14 \quad \text{(Equation 4)}\]
For equation (3):
\[x - 2(-3 - 2z) + 2z = -2\]
\[x + 6 + 4z + 2z = -2\]
\[x + 6z = -8 \quad \text{(Equation 5)}\]
- Solving for the Variables
Now we have a smaller system of two equations:
4) \(x - 5z = 14\)
5) \(x + 6z = -8\)
Subtracting Equation (4) from Equation (5) allows us to eliminate \(x\):
\[(x + 6z) - (x - 5z) = -8 - 14\]
\[11z = -22\]
\[z = -2\]
With the value of \(z\) known, we can find \(x\) using Equation (5):
\[x + 6(-2) = -8\]
\[x - 12 = -8\]
\[x = 4\]
Finally, we substitute \(z = -2\) back into our expression for \(y\):
\[y = -3 - 2(-2)\]
\[y = -3 + 4\]
\[y = 1\]
- Verification and Final Conclusion
We have found the values \(x=4\), \(y=1\), and \(z=-2\). Let's verify these values with the original system:
1) \(0(4) + 1(1) + 2(-2) = 1 - 4 = -3\) (Correct)
2) \(1(4) + 2(1) - (-2) = 4 + 2 + 2 = 8\) (Correct)
3) \(1(4) - 2(1) + 2(-2) = 4 - 2 - 4 = -2\) (Correct)
The solution to the system of linear equations is \((x, y, z) = (4, 1, -2)\). The final answer is \(\boxed{{(4, 1, -2)}}\).
Examples
In physics and engineering, systems of linear equations are frequently used to analyze circuits or structural forces. For example, when calculating the currents \(I_1, I_2, I_3\) through various branches of a circuit, Kirchhoff's laws provide a set of linear equations in the form of a matrix. Solving this matrix, just as we solved for \(x, y, z\), allows engineers to determine the exact flow of electricity in complex systems, ensuring components receive the correct power levels to operate safely and efficiently.