Q 269

Question

Use Cramer’s Rule to Solve Systems of Equations.

In the following exercises, solve each system of equations using Cramer’s Rule.

$\{\begin{cases} 2 x + 3 y + z = 12 \\ x + y + z = 9 \\ 3 x + 4 y + 2 z = 20 \end{cases}$

Step-by-Step Solution

Verified

Answer

The system is inconsistent and it has no solution.

1Step 1 Given system of equations are,

$\{\begin{cases} 2 x + 3 y + z = 12 \\ x + y + z = 9 \\ 3 x + 4 y + 2 z = 20 \end{cases}$

2Step 2 to find:

We need to find the solution for the system of equations by using the Cramer's rule.

3Step 3 Find the determinant D by using the coefficients of the variables.

$D = |\begin{matrix} 2 & 3 & 1 \\ 1 & 1 & 1 \\ 3 & 4 & 2 \end{matrix}| = 2 (2 - 4) - 3 (2 - 3) + 1 (4 - 3) = 2 (- 2) - 3 (- 1) + 1 (1) = - 4 + 3 + 1 = 0$

We cannot use the Cramer's rule to solve this system.

But by looking at the value of the determinants $D_{x}, D_{y}, D_{z}$ , we can determine whether the system is dependent or inconsistent.

4Step 4 Evaluate the determinant D x , use the constants to replace the coefficients of x .

$D_{x} = |\begin{matrix} 12 & 3 & 1 \\ 9 & 1 & 1 \\ 20 & 4 & 2 \end{matrix}| = 12 (2 - 4) - 3 (18 - 20) + 1 (36 - 20) = 12 (- 2) - 3 (- 2) + 1 (16) = - 24 + 6 + 16 = - 2$

5Step 5 Evaluate the determinant D y ,use the constants to replace the coefficients of y .

$D_{y} = |\begin{matrix} 2 & 12 & 1 \\ 1 & 9 & 1 \\ 3 & 20 & 2 \end{matrix}| = 2 (18 - 20) - 12 (2 - 3) + 1 (20 - 27) = 2 (- 2) - 12 (- 1) + 1 (- 7) = - 4 + 12 - 7 = 1$

6Step 6 Evaluating the determinant D z , by using the constants to replace the coefficients of z .

$D_{z} = |\begin{matrix} 2 & 3 & 12 \\ 1 & 1 & 9 \\ 3 & 4 & 20 \end{matrix}| = 2 (20 - 36) - 3 (20 - 27) + 12 (4 - 3) = 2 (- 16) - 3 (- 7) + 12 (1) = - 32 + 21 + 12 = 1$

All the determinants are not equal to zero.

So, the system is inconsistent and it has no solution.

Q 268

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