- Express \(z\) in terms of \(y\) using the second equation: \(z = 27 - 6y\).
- Substitute \(z\) into the first and third equations to reduce the system to two variables (\(x, y\)).
- Solve the resulting system of two equations to find \(y = 5\) and \(x = 7\).
- Substitute \(y = 5\) back into the expression for \(z\) to find \(z = -3\).
- The final solution is \(\boxed{(7, 5, -3)}\).
Explanation
- Analysis of the System
We are given a system of three linear equations with three variables \(x, y,\) and \(z\):
1) \(x - 3y - 5z = 7\)
2) \(6y + z = 27\)
3) \(6x + 5y - 6z = 85\)
Our objective is to find the unique ordered triple \((x, y, z)\) that satisfies all three equations simultaneously. We can achieve this by using the method of substitution to reduce the system step-by-step.
- Reducing the System
First, we look at the second equation, \(6y + z = 27\), which is convenient to solve for \(z\) in terms of \(y\):
\[z = 27 - 6y\]
Next, we substitute this expression for \(z\) into the first equation:
\[x - 3y - 5(27 - 6y) = 7\]
\[x - 3y - 135 + 30y = 7\]
\[x + 27y = 142 \quad (\text{Equation } 1')\]
Now, we substitute \(z = 27 - 6y\) into the third equation:
\[6x + 5y - 6(27 - 6y) = 85\]
\[6x + 5y - 162 + 36y = 85\]
\[6x + 41y = 247 \quad (\text{Equation } 3')\]
- Solving for y
We now have a system of two equations with two variables:
1') \(x + 27y = 142\)
3') \(6x + 41y = 247\)
From equation (1'), we can express \(x\) in terms of \(y\):
\[x = 142 - 27y\]
Substitute this expression for \(x\) into equation (3'):
\[6(142 - 27y) + 41y = 247\]
\[852 - 162y + 41y = 247\]
\[852 - 121y = 247\]
\[121y = 852 - 247\]
\[121y = 605\]
\[y = \frac{605}{121} = 5\]
- Final Solution and Verification
With \(y = 5\) found, we can determine \(x\) using the relation \(x = 142 - 27y\):
\[x = 142 - 27(5) = 142 - 135 = 7\]
Finally, substitute \(y = 5\) into the expression for \(z\):
\[z = 27 - 6(5) = 27 - 30 = -3\]
Thus, the solution to the system is \(x = 7, y = 5,\) and \(z = -3\). We check the result in the original equations:
1) \(7 - 3(5) - 5(-3) = 7 - 15 + 15 = 7\)
2) \(6(5) + (-3) = 30 - 3 = 27\)
3) \(6(7) + 5(5) - 6(-3) = 42 + 25 + 18 = 85\)
All equations are satisfied, confirming the triple \((7, 5, -3)\).
Examples
Systems of linear equations are used extensively in economics, engineering, and logistics to find optimal values. For example, if a small business owner has a fixed budget and sells three types of products, they can set up a system of equations based on resource constraints (like raw materials or labor hours) to determine how many units of each product they should produce to exactly exhaust their resources and maximize their revenue. Solving these systems ensures efficient resource allocation.