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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. 47 (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.
$\{\begin{cases} x - y = 6 \\ 2 x - 3 z = 16 \\ 2 y + z = 4 \end{cases}$

Short Answer

Expert verified

The solution of the system is $x = 8, y = 2, z = 0$ ,or, using ordered triplets, $(8, 2, 0)$ .

Step by step solution

01

Step 1. Given information.

The given system of equation are

$\{\begin{cases} x - y = 6 \\ 2 x - 3 z = 16 \\ 2 y + z = 4 \end{cases}$

02

Step 2. Calculation.

The augmented matrix of the system is:

$[\begin{matrix} \begin{matrix} 1 & - 1 & 0 \\ 2 & 0 & - 3 \\ 0 & 2 & 1 \end{matrix} & \begin{matrix} 6 \\ 16 \\ 4 \end{matrix} \end{matrix}]$

Perform the row operations $R_{2} = r_{2} - 2 r_{1}$ :

$[\begin{matrix} \begin{matrix} 1 & - 1 & 0 \\ 0 & 2 & - 3 \\ 0 & 2 & 1 \end{matrix} & \begin{matrix} 6 \\ 4 \\ 4 \end{matrix} \end{matrix}]$

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

$[\begin{matrix} \begin{matrix} 1 & - 1 & 0 \\ 0 & 2 & - 3 \\ 0 & 0 & 4 \end{matrix} & \begin{matrix} 6 \\ 4 \\ 0 \end{matrix} \end{matrix}]$

Use the obtained matrix to write the system of equations.

$x - y = 6 2 y - 3 z = 4 4 z = 0$

03

Step 3. Solve the equation.

Solve the equations to find the solution set.

$4 z = 0 z = 0$

Substitute $z = 0$ into the second equation.

$2 y - 3 z = 4 2 y - 3 (0) = 4 2 y = 4 y = 2$

Substitute $y = 2$ into the first equation.

$x - y = 6 x - 2 = 6 x = 6 + 2 x = 8$

Therefore, the solution of the system is $x = 8, y = 2, z = 0$ or, using ordered triplets, $(8, 2, 0)$ .

04

Step 4. Conclusion.

The solution of the system is $x = 8, y = 2, z = 0$ or, using ordered triplets, $(8, 2, 0)$ .

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