Consider the system { x+2 y+6 z =5 -3 x-6 y+5 z =8 2 x+6 y+9 z =7 . a. Verify that

step 1 So in this question, it is already given to me that x equal to minus 1, y is equal to 0, and z i

Consider the system { x+2 y+6 z =5 -3 x-6 y+5 z =8 ...

Consider the system $$\left\{\begin{aligned} x+2 y+6 z &=5 \\ -3 x-6 y+5 z &=8 \\ 2 x+6 y+9 z &=7 \end{aligned}\right.$$ a. Verify that $x=-1, y=0, z=1$ is a solution of the system. b. Find the determinant of the coefficient matrix. c. Without solving the system, determine whether there are any other tions. d. Can Cramer's Rule be used to solve this system? Why or why not?

Consider the system $$\left\{\begin{aligned}
x+2 y+6 z &=5 \\
-3 x-6 y+5 z &=8 \\
2 x+6 y+9 z &=7
\end{aligned}\right.$$ a. Verify that $x=-1, y=0, z=1$ is a solution of the system.
b. Find the determinant of the coefficient matrix.
c. Without solving the system, determine whether there are any other tions.
d. Can Cramer's Rule be used to solve this system? Why or why not?

Key Concepts

System of Linear Equations

A system of linear equations is a collection of one or more linear equations involving the same set of variables. The goal is to find values for the variables that satisfy all the equations simultaneously. Such systems can be represented in various forms, including as augmented matrices or through algebraic equations, which facilitates analysis using matrix operations and other algebraic techniques.

Verifying Solutions

Verifying a solution of a system of linear equations involves substituting the proposed values into each equation to ensure that all equations are satisfied. This process confirms the correctness of the solution and is a useful step in both homework problems and checking work in applied settings.

Coefficient Matrix

The coefficient matrix is constructed by taking the coefficients of the variables from each linear equation in the system. It is a key component in matrix representations of linear systems and is crucial in determining properties such as the system’s solvability and the uniqueness of its solutions.

Determinant

The determinant is a scalar value that can be computed from the elements of a square matrix and provides important information about the matrix, including whether it is invertible. In the context of systems of linear equations, a nonzero determinant indicates that the system has a unique solution, while a zero determinant suggests that the system may have either no solutions or infinitely many.

Cramer’s Rule

Cramer’s Rule is a method used to solve systems of linear equations with as many equations as unknowns, provided that the determinant of the coefficient matrix is nonzero. It involves replacing one column of the coefficient matrix with the right-hand side values and computing the determinant of the new matrix, with each variable calculated as the ratio of this determinant to the original determinant.

Uniqueness and Existence of Solutions

The existence and uniqueness of solutions in a system of linear equations are tightly linked to the determinant of the coefficient matrix. If the determinant is nonzero, the system has a unique solution; however, if it is zero, the system may be underdetermined or inconsistent. Techniques, such as analyzing the rank of the matrix, are used in these cases to determine whether any solutions exist.

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