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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 222 (page 444)

Solve each system of equations using a matrix.
$\{\begin{cases} 2 x + 3 y + z = 12 \\ x + y + z = 9 \\ 3 x + 4 y + 2 z = 20 \end{cases}$

Short Answer

Expert verified

The system of linear equations doesn't have any solution.

Step by step solution

01

Step 1. Given information.

Consider the given system of equations,

$\{\begin{cases} 2 x + 3 y + z = 12 \\ x + y + z = 9 \\ 3 x + 4 y + 2 z = 20 \end{cases}$

02

Step 2. Write in augmented form.

The augmented matrix for the given system of equations is

$[\begin{array}{cccr} 2 & 3 & 1 & 12 \\ 1 & 1 & 1 & 9 \\ 3 & 4 & 2 & 20 \end{array}]$

03

Step 3. Apply row operations.

Apply $\frac{R_{1}}{2} 鈫� R_{1}$ and $\frac{R_{3}}{- 1} 鈫� R_{3}$ ,

$[\begin{array}{cccr} 1 & \frac{3}{2} & \frac{1}{2} & 6 \\ 0 & - \frac{1}{2} & \frac{1}{2} & 3 \\ 3 & 4 & 2 & 20 \end{array}]$

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

$[\begin{array}{cccr} 1 & \frac{3}{2} & \frac{1}{2} & 6 \\ 0 & 1 & - 1 & - 6 \\ 0 & - \frac{1}{2} & \frac{1}{2} & 2 \end{array}]$

Apply $R_{3} + \frac{1}{2} 脳 R_{2} 鈫� R_{3}$ ,

$[\begin{array}{cccr} 1 & \frac{3}{2} & \frac{1}{2} & 6 \\ 0 & 1 & - 1 & - 6 \\ 0 & 0 & 0 & - 1 \end{array}]$

Apply $\frac{R_{3}}{- 1} 鈫� R_{3}$ ,

$[\begin{array}{cccr} 1 & \frac{3}{2} & \frac{1}{2} & 6 \\ 0 & 1 & - 1 & - 6 \\ 0 & 0 & 0 & 1 \end{array}]$

Now, the matrix is in row-echelon form.

04

Step 4. Write in system of equations.

Writing the corresponding system of equations,

$x + \frac{3}{2} y + \frac{1}{2} z = 6 . . . . . . (i) y - z = - 6 . . . . . . (i i) 0 = 1 . . . . . . (i i i)$

As equation (iii) is a false statement.

Therefore, it is not possible to solve and is an inconsistent system.

Hence, the system of linear equations has no solution.

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Most popular questions from this chapter

Solve the systems of equations by substitution.

$\{\begin{cases} 2 x + y = - 4 \\ 3 x - 2 y = - 6 \end{cases}$

Translate to a system of equation and solve.

The difference of two supplementary angles is $8$ degrees. Find the measures of the angles.

The sum of two number is $30$ . One number is $4$ less than the other. Find the numbers.

To decide whether it would be more convenient to solve the system of equations by substitution or

elimination.

$(a) \{\begin{cases} 14 x - 15 y = - 30 \\ 7 x + 2 y = 10 \end{cases} (b) \{\begin{cases} x = 9 y - 11 \\ 2 x - 7 y = - 27 \end{cases}$

Solve the systems of equations by substitution.

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

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