Solve the following triangular linear system:
[tex]
\begin{aligned}
x-5 y+3 z & = -8 \\
y-2 z & = -8 \\
z & = 6
\end{aligned}
[/tex]
Answer (2)
Substitute z = 6 into the second equation to find y = 4 .
Substitute y = 4 and z = 6 into the first equation to find x = − 6 .
The solution to the system is ( − 6 , 4 , 6 ) .
Explanation
Understanding the Problem We are given a triangular linear system of equations:
Equation 1: x − 5 y + 3 z = − 8 Equation 2: y − 2 z = − 8 Equation 3: z = 6
Our goal is to solve this system for x , y , and z . We can use back-substitution because the system is triangular.
Solving for y First, we substitute the value of z from Equation 3 into Equation 2 to solve for y :
y − 2 ( 6 ) = − 8 y − 12 = − 8 y = − 8 + 12 y = 4
Solving for x Now that we have the values of y and z , we can substitute them into Equation 1 to solve for x :
x − 5 ( 4 ) + 3 ( 6 ) = − 8 x − 20 + 18 = − 8 x − 2 = − 8 x = − 8 + 2 x = − 6
The Solution Therefore, the solution to the system of equations is the ordered triple ( − 6 , 4 , 6 ) .
Examples
Triangular linear systems are used in various fields such as electrical engineering (circuit analysis), computer graphics (transformations), and economics (input-output models). For instance, in circuit analysis, you might have a system of equations representing the currents and voltages in different parts of a circuit. If the system is triangular, you can easily solve for the unknowns using back-substitution, which simplifies the analysis and design of the circuit.
To solve the triangular linear system, we substitute z = 6 into the second equation to find y = 4 , then substitute both y and z into the first equation to find x = − 6 . Thus, the solution is ( − 6 , 4 , 6 ) .
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