Question

Determine whether the two systems of linear equations yield the same solution. If so, find the solution using matrices. (a) \(\left\{$$\begin{array}{rl}x-4y+5z& =27\\ y-7z& =-54\\ z& =8\end{array}$$\right.\) (b) \(\left\{$$\begin{array}{rl}x-6y+z& =15\\ y+5z& =42\\ z& =8\end{array}$$\right.\)

Short answer

Yes, the two systems of equations have the same solutions, which are $x=113$, $y=-16$, and $z=8$.

Step-by-step solution

Convert to Matrix Form

Convert the two systems of equations into matrices. The terms on the left-hand side of the equal sign represent the coefficients in the matrix. The terms on the right-hand side of the equal sign, which are solution of each equation, will be put into a separate matrix. So, the first system of equations becomes matrix A: $$\begin{array}{cccccccccc}1& -4& 5& 270& 1& -7& -540& 0& 1& 8\end{array}$$, and the second one becomes matrix B: $$\begin{array}{cccccccccc}1& -6& 1& 150& 1& 5& 420& 0& 1& 8\end{array}$$.

Solve the Matrices

Now solve the matrices independently to find the values for variables x, y, and z. For matrix A, since it's already in Row-Echelon form, start from the bottom to solve for z, then substitute z into the second equation to solve for y, and finally substitute y and z into the first equation to get x. Thus, $x=113$, $y=-16$, and $z=8$. For matrix B, proceed with the same method for solving the matrix. Here too, ($x=113$, $y=-16$, $z=8$).

Compare Solutions

The solution for both system of equations are identical. Thus, the two systems yield the same solution.

Conclusion

The analysis of both system of linear equations confirms that they yield the same solution, which is $x=113$, $y=-16$, and $z=8$.