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" $[\begin{matrix} - 3 & 1 & 1 & - 4 \\ - 1 & 2 & - 2 & 1 \\ 2 & - 1 & - 1 & - 1 \end{matrix}] \overset{R_{1} \leftrightarrow R_{2}}{\to} [\begin{matrix} - 1 & 2 & - 2 & 1 \\ - 3 & 1 & 1 & - 4 \\ 2 & - 1 & - 1 & - 1 \end{matrix}] \overset{- 3 R_{1} + R_{2}}{\to} [\begin{matrix} - 1 & 2 & - 2 & 1 \\ 0 & - 5 & 7 & - 7 \\ 2 & - 1 & - 1 & - 1 \end{matrix}] \overset{2 R_{1} + R_{3}}{\to} [\begin{matrix} - 1 & 2 & - 2 & 1 \\ 0 & - 5 & 7 & - 7 \\ 0 & - 4 & - 5 & 1 \end{matrix}] \overset{R_{3} \to 5 R_{2} + 7 R_{3}}{\to} [\begin{matrix} - 1 & 2 & - 2 & 1 \\ 0 & - 5 & 7 & - 7 \\ 0 & - 4 & 0 & - 28 \end{matrix}] \overset{R_{3} \to \frac{1}{4} R_{3}}{\to} [\begin{matrix} - 1 & 2 & - 2 & 1 \\ 0 & - 5 & 7 & - 7 \\ 0 & 1 & 0 & 7 \end{matrix}]$

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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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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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

Logic and Functions

Theoretical and Mathematical Physics

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Study anywhere. Anytime. Across all devices.

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