Solve the $3 \times 3$ system shown below. Enter the values of $x, y$, and $z$.
$\begin{array}{l}
x+2 y-z=-3 \\
2 x-y+z=5 \\
x-y+z=4 \\
x=\square \\
y=\square
\end{array}$
Answer (1)
Add the first two equations to eliminate z , resulting in 3 x + y = 2 .
Subtract the third equation from the second to eliminate z , giving x = 1 .
Substitute x = 1 into 3 x + y = 2 to find y = − 1 .
Substitute x = 1 and y = − 1 into the third equation to solve for z = 2 .
The solution is x = 1 , y = − 1 , z = 2 .
Explanation
Understanding the Problem We have a system of three linear equations with three unknowns: x, y, and z. The equations are:
x + 2y - z = -3
2x - y + z = 5
x - y + z = 4 We need to find the values of x, y, and z that satisfy all three equations simultaneously.
Eliminating z from Equations (1) and (2) Add equation (1) and equation (2) to eliminate z: ( x + 2 y − z ) + ( 2 x − y + z ) = − 3 + 5 This simplifies to: 3 x + y = 2 Call this equation (4).
Eliminating z from Equations (2) and (3) Subtract equation (3) from equation (2) to eliminate z: ( 2 x − y + z ) − ( x − y + z ) = 5 − 4 This simplifies to: x = 1
Solving for y Substitute x = 1 into equation (4): 3 ( 1 ) + y = 2 This gives: y = 2 − 3 = − 1
Solving for z Substitute x = 1 and y = -1 into equation (3): 1 − ( − 1 ) + z = 4 This gives: 1 + 1 + z = 4 So, z = 4 − 2 = 2
Final Answer Therefore, the solution is: x = 1 , y = − 1 , z = 2 Thus, the values of x, y, and z are 1, -1, and 2, respectively.
Examples
Systems of equations are used in various fields, such as engineering, economics, and computer science. For example, in electrical engineering, systems of equations can be used to analyze circuits and determine the current and voltage at different points in the circuit. In economics, systems of equations can be used to model supply and demand curves and determine the equilibrium price and quantity. Understanding how to solve systems of equations is a fundamental skill that can be applied to many real-world problems.
