Question

In Exercises 45–48, explain why the system of equations cannot be solved using Cramer's Rule. Then use Gaussian elimination to solve the system.

Accepted Answer

Step 1: Write down the coefficient matrix of the system:

$$
A = \begin{bmatrix} 4 & -3 & -2 \ 8 & -6 & -4 \end{bmatrix}
$$

Note that this is a 2x3 matrix because there are 2 equations and 3 variables (x, y, z). Step 2: Explain why Cramer's Rule cannot be used:

Cramer's Rule requires a square coefficient matrix (same number of equations as variables) and a non-zero determinant. Here, the coefficient matrix is not square (2x3), so the determinant is not defined. Therefore, Cramer's Rule cannot be applied. Step 3: Set up the augmented matrix for Gaussian elimination:

$$
\left[ \begin{array}{ccc|c}
4 & -3 & -2 & 12 \
8 & -6 & -4 & 22
\end{array} ight]
$$

This matrix includes the coefficients and the constants from the right side of the equations. Step 4: Use row operations to simplify the augmented matrix:

- Multiply the first row by 2 and subtract it from the second row to eliminate the x-term in the second row:

Row2 = Row2 - 2 * Row1

This will give a new second row. Step 5: Analyze the resulting system:

- If the second row becomes a row of zeros with a non-zero constant, the system is inconsistent (no solution).
- If the second row becomes all zeros including the constant, the system has infinitely many solutions.
- Otherwise, continue with back substitution to find the values of variables.

In Exercises 45–48, explain why the system of equations cannot be solved using Cramer's Rule. Then use Gaussian elimination to solve the system.

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Concetti chiave

Cramer's Rule and Determinants

Gaussian Elimination

System Consistency and Dependency