- Use equation (3) to define \(x = y - z\).
- Substitute \(x = y - z\) into equations (1) and (2) to obtain a system of two equations in \(y\) and \(z\): \(3y + z = 5\) and \(9y + 2z = 19\).
- Solve the resulting system for \(y\) and \(z\) to find \(y = 3\) and \(z = -4\).
- Calculate \(x = y - z = 3 - (-4) = 7\).
- The solution to the system is \(\boxed{(7, 3, -4)}\).
Explanation
- Analysis and Setup
To solve the system of linear equations, we first analyze the three given equations:
1) \(-3x + 6y - 2z = 5\)
2) \(7x + 2y + 9z = 19\)
3) \(-x + y - z = 0\)
This system describes three planes in 3D space, and we seek the point \((x, y, z)\) where these planes intersect. We will use the substitution method, starting with the simplest equation to reduce the number of variables.
- Substituting and Simplifying the System
We start by using the third equation to express \(x\) in terms of \(y\) and \(z\):
\(-x + y - z = 0 \implies x = y - z\)
Now, we substitute this expression for \(x\) into the first equation:
\(-3(y - z) + 6y - 2z = 5\)
\(-3y + 3z + 6y - 2z = 5\)
\(3y + z = 5\) (Equation A)
Next, we substitute the expression for \(x\) into the second equation:
\(7(y - z) + 2y + 9z = 19\)
\(7y - 7z + 2y + 9z = 19\)
\(9y + 2z = 19\) (Equation B)
- Solving for Variables
Now we have a smaller system of two linear equations with two variables:
A\(3y + z = 5\)
B\(9y + 2z = 19\)
From Equation A, we can isolate \(z\):
\(z = 5 - 3y\)
Substitute this into Equation B:
\(9y + 2(5 - 3y) = 19\)
\(9y + 10 - 6y = 19\)
\(3y + 10 = 19\)
\(3y = 9 \implies y = 3\)
Now, find \(z\) using \(z = 5 - 3y\):
\(z = 5 - 3(3) = 5 - 9 = -4\)
Finally, find \(x\) using \(x = y - z\):
\(x = 3 - (-4) = 7\)
- Verification and Final Solution
We have found the values \(x = 7\), \(y = 3\), and \(z = -4\). To ensure this solution is correct, we verify it against the original equations:
1) \(-3(7) + 6(3) - 2(-4) = -21 + 18 + 8 = 5\) (Checks out)
2) \(7(7) + 2(3) + 9(-4) = 49 + 6 - 36 = 19\) (Checks out)
3) \(-7 + 3 - (-4) = -7 + 3 + 4 = 0\) (Checks out)
All equations are satisfied. The solution is the ordered triple \((7, 3, -4)\).
Examples
This system of equations is a fundamental tool for solving problems in fields such as engineering, economics, and physics. For example, if you are determining the currents in a complex circuit using Kirchhoff's laws, or balancing a chemical equation, you will arrive at a system of linear equations just like this one. Mastering these techniques allows for solving for multiple unknowns in any situation where linear relationships between variables are known.