- Use the subtraction of equation (1) from (2) to isolate \(y\), yielding \(y=4\).
- Substitute \(y=4\) into the sum of equations (1) and (3) to isolate \(x\), yielding \(x=9\).
- Substitute \(x=9\) and \(y=4\) into equation (1) to solve for \(z\), yielding \(z=3\).
- Conclude that the unique solution is \((x, y, z) = \boxed{(9, 4, 3)}\).
Explanation
- Problem Analysis and Objective
We are given the following system of linear equations in three variables \(x, y,\) and \(z\):
1) \(x - 2y + z = 4\)
2) \(x + 3y + z = 24\)
3) \(3x - y - z = 20\)
Our objective is to solve for \(x, y,\) and \(z\) using the addition method (also known as the elimination method). This method involves combining the equations to systematically eliminate variables, making the system easier to solve. We have three variables, so our plan is to reduce the system step-by-step until we find the value of each variable.
- Eliminating \(z\) to Find \(y\)
To eliminate the variable \(z\), we can observe the coefficients of \(z\) in our equations. In equations (1) and (2), the coefficient of \(z\) is 1. We can subtract equation (1) from equation (2):
\((x + 3y + z) - (x - 2y + z) = 24 - 4\)
This simplifies to:
\(x - x + 3y - (-2y) + z - z = 20\)
\(5y = 20\)
Dividing both sides by 5, we find the value of \(y\):
\(y = 4\)
- Solving for \(x\)
Now that we have \(y = 4\), we can eliminate \(z\) using a different pair of equations. Let's add equations (1) and (3) since the coefficients of \(z\) are 1 and -1, respectively:
\((x - 2y + z) + (3x - y - z) = 4 + 20\)
This simplifies to:
\(4x - 3y = 24\)
Now, substitute the value \(y = 4\) into this new equation to solve for \(x\):
\(4x - 3(4) = 24\)
\(4x - 12 = 24\)
\(4x = 36\)
\(x = 9\)
- Solving for \(z\)
We have found \(x = 9\) and \(y = 4\). To find \(z\), we can substitute these values into any of the original equations. Using equation (1):
\(x - 2y + z = 4\)
\(9 - 2(4) + z = 4\)
\(9 - 8 + z = 4\)
\(1 + z = 4\)
\(z = 3\)
- Verification and Conclusion
Finally, we check our solution \((x, y, z) = (9, 4, 3)\) against all original equations to ensure consistency:
1) \(9 - 2(4) + 3 = 9 - 8 + 3 = 4\) (Correct)
2) \(9 + 3(4) + 3 = 9 + 12 + 3 = 24\) (Correct)
3) \(3(9) - (4) - 3 = 27 - 4 - 3 = 20\) (Correct)
The values satisfy all given equations. Thus, the solution to the system is \((x, y, z) = (9, 4, 3)\).
Examples
This method is essential in various fields such as engineering and finance where one must solve systems of constraints to find optimal values. For instance, if you have a budget of \(\$24\) and need to purchase three different items \(x, y, z\) with specific price weights, the system of equations represents your spending constraints. Solving this ensures you find the unique combination of items that satisfy all your shopping requirements simultaneously.