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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. 53 (page 730)

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.
$\{\begin{cases} - x + y + z = - 1 \\ - x + 2 y - 3 z = - 4 \\ 3 x - 2 y - 7 z = 0 \end{cases}$

Short Answer

Expert verified

The solution of the system is $x = 5 z - 2, y = 4 z - 3$ where, z is any real number.

Step by step solution

01

Step 1. Given information.

The given system of equation are

$\{\begin{cases} - x + y + z = - 1 \\ - x + 2 y - 3 z = - 4 \\ 3 x - 2 y - 7 z = 0 \end{cases}$

02

Step 2. Calculation.

The augmented matrix of the system is:

$[\begin{matrix} \begin{matrix} - 1 & 1 & 1 \\ - 1 & 2 & - 3 \\ 3 & - 2 & - 7 \end{matrix} & \begin{matrix} - 1 \\ - 4 \\ 0 \end{matrix} \end{matrix}]$

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

$[\begin{matrix} \begin{matrix} - 1 & 1 & 1 \\ 0 & 1 & - 4 \\ 0 & 4 & - 16 \end{matrix} & \begin{matrix} - 1 \\ - 3 \\ - 12 \end{matrix} \end{matrix}]$

Perform the row operations $R_{3} = r_{3} - 4 r_{2}$ :

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

The above matrix is in reduced row echelon form. The corresponding system of equations is

localid="1647511124810" $- x + y + z = - 1 (1) y - 4 z = - 3 鈬� y = 4 z - 3 (2)$

03

Step 3. Solve the equation.

Substitute $y = 4 z - 3$ into equation (1).

$- x + 4 z - 3 + z = - 1 - x + 5 z = 2 x = 5 z - 2$

Therefore, the solution of the system is $x = 5 z - 2, y = 4 z - 3$ where, z is any real number.

04

Step 4. Conclusion.

The solution of the system is $x = 5 z - 2, y = 4 z - 3$ where, z is any real number.

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Most popular questions from this chapter

Verify that the values of the variables listed are solutions of the system of equations.

$\{\begin{matrix} 2 x - y = 5 \\ 5 x + 2 y = 8 \end{matrix} x = 2, y = - 1; (2, - 1)$

In Problems 5鈥�24, graph each equation of the system. Then solve the system to find the points of intersection.

$y = \sqrt{x}; y = 6 - x$

Fencing: A farmer has 300 feet of fence available to enclose a 4500-square-foot region in the shape of adjoining squares, with sides of length x and y. See the figure. Findx and y.

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$x y = 1; y = 2 x + 1$

If the lines that make up a system of two linear equations are coincident, then the system is __________ and the equations are ___________.

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