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Intermediate Algebra

OPENSTAX

$Math Studyset 91影视 Explanations$ Math

OER 2017 Edition 路 1346 pages

ISBN: 9780998625720

Chapter 4: Q. 4.88 (page 442)

Solve the system of equations using a matrix: $\{\begin{cases} x - y - z = 1 \\ - x + 2 y - 3 z = - 4 \\ 3 x - 2 y - 7 z = 0 \end{cases}$

Short Answer

Expert verified

The solution for the system of linear equation is $(5 z - 2, 4 z - 3, z)$ where, $z$ is any real number.

Step by step solution

01

Step 1. Given information.

Consider the given system of equations,

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

02

Step 2. Write in augmented form.

The augmented matrix for the given system of equations is

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

03

Step 3. Apply row operations.

Apply $R_{2} + R_{1} 鈫� R_{2}$ and $R_{3} - 3 脳 R_{1} 鈫� R_{3}$ ,

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

Apply $R_{3} - R_{2} 鈫� R_{3}$ ,

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

04

Step 4. Write in system of equations.

Writing the corresponding system of equations,

$x - y - z = 1 . . . . . . (i) y - 4 z = - 3 . . . . . . (i i)$

Solve equation (ii) for $y$ ,

$y = 4 z - 3$

Substitute the value of $y$ in equation (i),

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

05

Step 5. Check the answers.

Substitute the values in equation (i),

$(5 z - 2) - (4 z - 3) - z = 1 5 z - 2 - 4 z + 3 - z = 1 1 = 1$

This is true.

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Most popular questions from this chapter

Determine whether the ordered pair is a solution to the system $\{\begin{cases} 3 x + y = 0 \\ x + 2 y = - 5 \end{cases}$

$(1, - 3)$
$(0, 0)$

Without graphing, determine the number of solutions and then classify the system of equations.

$(a) \{\begin{cases} y = \frac{1}{3} x - 5 \\ x - 3 y = 6 \end{cases} (b) \{\begin{cases} x + 4 y = 12 \\ - x + y = 3 \end{cases}$

Solve the system of equations by substitution

$\{\begin{cases} 4 x + y = 10 \\ x - 2 y = - 20 \end{cases}$

Without graphing, determine the number of solutions and then classify the system of equations.

$\{\begin{cases} 5 x 鈭� 2 y = 10 \\ y = \frac{5}{2} x - 5 \end{cases}$

Solve the system by graphing: $\{\begin{cases} y = \frac{1}{2} x - 4 \\ 2 x - 4 y = 16 \end{cases}$

See all solutions

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