Q 218 Solve each system of equations u... [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 218 (page 444)

Solve each system of equations using a matrix.
$\{\begin{cases} x + 2 z = 0 \\ 4 y + 3 z = - 2 \\ 2 x - 5 y = 3 \end{cases}$

Short Answer

Expert verified

The solution for the system of linear equation is $(4, 1, - 2)$ .

Step by step solution

Step 1. Given information.

Consider the given system of equations,

$\{\begin{cases} x + 2 z = 0 \\ 4 y + 3 z = - 2 \\ 2 x - 5 y = 3 \end{cases}$

Step 2. Write in augmented form.

The augmented matrix for the given system of equations is

$[\begin{array}{r} 1 & 0 & 2 & 0 \\ 0 & 4 & 3 & - 2 \\ 2 & - 5 & 0 & 3 \end{array}]$

Step 3. Apply row operations.

Apply $R_{3} - 2 脳 R_{1} 鈫� R_{3}$ ,

$[\begin{array}{lllr} 1 & 0 & 2 & 0 \\ 0 & 4 & 3 & - 2 \\ 0 & - 5 & - 4 & 3 \end{array}]$

Apply $\frac{R_{2}}{4} 鈫� R_{2}$ ,

$[\begin{array}{lllc} 1 & 0 & 2 & 0 \\ 0 & 1 & \frac{3}{4} & - \frac{1}{2} \\ 0 & - 5 & - 4 & 3 \end{array}]$

Apply $R_{3} + 5 脳 R_{2} 鈫� R_{3}$ ,

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

Apply $- 4 脳 R_{3} 鈫� R_{3}$ ,

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

Now, the matrix is in row-echelon form.

Step 4. Write in system of equations.

Writing the corresponding system of equations,

$x + 2 z = 0 . . . . . . (i) y + \frac{3}{4} z = - \frac{1}{2} . . . . . . (i i) z = - 2 . . . . . . (i i i)$

Substitute $z = - 2$ in equation (i),

$x + 2 (- 2) = 0 x - 4 = 0 x = 4$

Substitute $z = - 2$ in equation (ii),

$y + \frac{3}{4} (- 2) = - \frac{1}{2} y - \frac{3}{2} = - \frac{1}{2} y = \frac{3}{2} - \frac{1}{2} y = \frac{2}{2} y = 1$

Step 5. Check the answers.

Substitute the values $x, z$ in equation (i),

$4 + 2 (- 2) = 0 4 - 4 = 0 0 = 0$

This is true.

Substitute the values $y, z$ in equation (ii),

$1 + \frac{3}{4} (- 2) = - \frac{1}{2} 1 - \frac{3}{2} = - \frac{1}{2} - \frac{1}{2} = - \frac{1}{2}$

This is also true.

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91影视

Solve each system of equations using a matrix.
$\{\begin{cases} x + 2 z = 0 \\ 4 y + 3 z = - 2 \\ 2 x - 5 y = 3 \end{cases}$

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Write in augmented form.

Step 3. Apply row operations.

Step 4. Write in system of equations.

Step 5. Check the answers.

One App. One Place for Learning.

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91影视

Solve each system of equations using a matrix.x+2z=04y+3z=-22x-5y=3

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Write in augmented form.

Step 3. Apply row operations.

Step 4. Write in system of equations.

Step 5. Check the answers.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

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Statistics

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Study anywhere. Anytime. Across all devices.

Solve each system of equations using a matrix.
$\{\begin{cases} x + 2 z = 0 \\ 4 y + 3 z = - 2 \\ 2 x - 5 y = 3 \end{cases}$