Q. 413 Solve the system of equations us... [FREE SOLUTION]

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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. 413 (page 485)

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

Short Answer

Expert verified

The solution of system of equation is $(5, 7, 4)$ .

Step by step solution

Step 1. Given information.

Given linear equations:

$\begin{array}{l} - 3 x + y + z = - 4 \\ - x + 2 y - 2 z = 1 \\ 2 x - y - z = - 1 \end{array}$

Step 2. Write the concept.

Let's write the equation in augmented form.

The augmented matrix is as follows:

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

Step 3. Calculation

Solve the augmented matrix

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

Let us interchange row 1 and row 2.

role="math" localid="1644679774023" $[\begin{matrix} - 3 & 1 & 1 & - 4 \\ - 1 & 2 & - 2 & 1 \\ 2 & - 1 & - 1 & - 1 \end{matrix}] \overset{R_{1} 鈫� R_{2}}{鈫�} [\begin{matrix} - 1 & 2 & - 2 & 1 \\ - 3 & 1 & 1 & - 4 \\ 2 & - 1 & - 1 & - 1 \end{matrix}] \overset{- 3 R_{1} + R_{2}}{鈫�} [\begin{matrix} - 1 & 2 & - 2 & 1 \\ 0 & - 5 & 7 & - 7 \\ 2 & - 1 & - 1 & - 1 \end{matrix}] \overset{2 R_{1} + R_{3}}{鈫�} [\begin{matrix} - 1 & 2 & - 2 & 1 \\ 0 & - 5 & 7 & - 7 \\ 0 & - 4 & - 5 & 1 \end{matrix}] \overset{R_{3} 鈫� 5 R_{2} + 7 R_{3}}{鈫�} [\begin{matrix} - 1 & 2 & - 2 & 1 \\ 0 & - 5 & 7 & - 7 \\ 0 & - 4 & 0 & - 28 \end{matrix}] \overset{R_{3} 鈫� \frac{1}{4} R_{3}}{鈫�} [\begin{matrix} - 1 & 2 & - 2 & 1 \\ 0 & - 5 & 7 & - 7 \\ 0 & 1 & 0 & 7 \end{matrix}]$

Step 4. Echelon form

Matrix in echelon form is

$[\begin{matrix} - 1 & 2 & - 2 & 1 \\ 0 & - 5 & 7 & - 7 \\ 0 & 1 & 0 & 7 \end{matrix}]$

Therefore the corresponding system of equation is

$- x + 2 y - 2 z = 1 - 5 y + 7 z = - 7 y = 7$

Step 5. Substitution method for remaining values.

Put $y = 7$ in the equation

$- 5 y + 7 z = - 7 - 5 (7) + 7 z = - 7 - 35 + 7 z = - 7 7 z = - 7 + 35 7 z = 28 z = 4$

Put $y = 7, z = 4$ in the equation

$- 3 x + y + z = - 4 - 3 x + 7 + 4 = - 4 - 3 x + 11 = - 4 - 3 x = - 15 x = 5$

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91影视

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

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Write the concept.

Step 3. Calculation

Step 4. Echelon form

Step 5. Substitution method for remaining values.

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91影视

Solve the system of equations using a matrix.-3x+y+z=-4-x+2y-2z=12x-y-z=-1

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Write the concept.

Step 3. Calculation

Step 4. Echelon form

Step 5. Substitution method for remaining values.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Decision Maths

Geometry

Discrete Mathematics

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Calculus

Study anywhere. Anytime. Across all devices.

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