Q. 48 In Problems 37鈥�72, solve each ... [FREE SOLUTION]

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Precalculus Enhanced with Graphing Utilities

Sullivan

$Math Studyset 91影视 Explanations$ Math

6th Edition 路 1200 pages

ISBN: 9780321795465

Chapter 11: Q. 48 (page 729)

In Problems 37鈥�72, solve each system of equations using matrices (row operations). If the system has no solution, say that it is inconsistent.

Short Answer

Expert verified

The solution of the system is $x = - 3, y = 2, z = 1$ ,or, using ordered triplets, $(- 3, 2, 1)$ .

Step by step solution

Step 1. Given information.

The given system of equation are

$\{\begin{cases} 2 x + y = - 4 \\ - 2 y + 4 z = 0 \\ 3 x - 2 z = - 11 \end{cases}$

Step 2. Calculation.

The augmented matrix of the system is:

$[\begin{matrix} \begin{matrix} 2 & 1 & 0 \\ 0 & - 2 & 4 \\ 3 & 0 & - 2 \end{matrix} & \begin{matrix} - 4 \\ 0 \\ - 11 \end{matrix} \end{matrix}]$

Perform the row operations $R_{3} = 2 r_{3} - 3 r_{2}$ :

role="math" $[\begin{matrix} \begin{matrix} 2 & 1 & 0 \\ 0 & - 2 & 4 \\ 0 & - 3 & - 4 \end{matrix} & \begin{matrix} - 4 \\ 0 \\ - 10 \end{matrix} \end{matrix}]$

Perform the row operations $R_{3} = 2 r_{3} - 3 r_{2}$ :

$[\begin{matrix} \begin{matrix} 2 & 1 & 0 \\ 0 & - 2 & 4 \\ 0 & 0 & - 20 \end{matrix} & \begin{matrix} - 4 \\ 0 \\ - 20 \end{matrix} \end{matrix}]$

Perform the row operations $R_{2} = - \frac{1}{2} r_{2}; R_{3} = - \frac{1}{20} r_{3}$ :

$[\begin{matrix} \begin{matrix} 2 & 1 & 0 \\ 0 & 1 & - 2 \\ 0 & 0 & 1 \end{matrix} & \begin{matrix} - 4 \\ 0 \\ 1 \end{matrix} \end{matrix}]$

Use the obtained matrix to write the system of equations.

$2 x + y = - 4 y - 2 z = 0 z = 1$

Step 3. Solve the equation.

Solve the equations to find the solution set.

Substitute $z = 1$ into the second equation.

$y - 2 z = 0 y - 2 (1) = 0 y - 2 = 0 y = 2$

Substitute $y = 2$ into the first equation.

$2 x + y = - 4 2 x + 2 = - 4 2 x = - 6 x = - \frac{6}{2} = - 3$

Therefore, the solution of the system is $x = - 3, y = 2, z = 1$ or, using ordered triplets, $(- 3, 2, 1)$ .

Step 4. Conclusion.

Therefore, the solution of the system is $x = - 3, y = 2, z = 1$ or, using ordered triplets, $(- 3, 2, 1)$ .

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91影视

In Problems 37鈥�72, solve each system of equations using matrices (row operations). If the system has no solution, say that it is inconsistent.

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Calculation.

Step 3. Solve the equation.

Step 4. Conclusion.

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