Chapter 11: Q. 21 (page 793)

In Problems 20–25, solve each system of equations using matrices. If the system has no solution, say that it is inconsistent.
$\{\begin{cases} 5 x - 6 y - 3 z = 6 \\ 4 x - 7 y - 2 z = - 3 \\ 3 x + y - 7 z = 1 \end{cases}$

Short Answer

Expert verified

The solutions to the given equation are $x = \frac{33}{4}, y = \frac{25}{6}, z = \frac{41}{12}$

Step by step solution

Step 1. Given information

The given system of equations is $\{\begin{cases} 5 x - 6 y - 3 z = 6 \\ 4 x - 7 y - 2 z = - 3 \\ 3 x + y - 7 z = 1 \end{cases}$

Step 2. Solve system of equations using matrix

$A = [\begin{array}{ccc} 5 & - 6 & - 3 \\ 4 & - 7 & - 2 \\ 3 & 1 & - 7 \end{array}], X = [\begin{array}{l} x \\ y \\ z \end{array}], B = [\begin{matrix} 6 \\ - 3 \\ 5 \end{matrix}]$

$A X = B X = A^{- 1} B$

The inverse of the matrix $A$ can now be determined by using the graphical utility.

So,

$A^{- 1} = [\begin{array}{ccc} \frac{17}{16} & - \frac{15}{16} & - \frac{3}{16} \\ \frac{11}{24} & - \frac{13}{24} & - \frac{1}{24} \\ \frac{25}{48} & - \frac{23}{48} & - \frac{11}{48} \end{array}]$

$X = A^{- 1} B \begin{aligned} X & = [\begin{array}{c} 6 \\ - 3 \\ 5 \end{array}] [\begin{array}{ccc} \frac{17}{16} & - \frac{15}{16} & - \frac{3}{16} \\ \frac{11}{24} & - \frac{13}{24} & - \frac{1}{24} \\ \frac{25}{48} & - \frac{23}{48} & - \frac{11}{48} \end{array}] \\ = [\begin{matrix} 6 \times \frac{17}{16} + (- 3) \times (- \frac{15}{16}) + 5 \times (- \frac{3}{16}) \\ 6 \times \frac{11}{24} + (- 3) \times (- \frac{13}{24}) + 5 \times (- \frac{1}{24}) \\ 6 \times \frac{25}{48} + (- 3) \times (- \frac{23}{48}) + 5 \times (- \frac{11}{48}) \end{matrix}] \\ = [\begin{matrix} \frac{102}{16} + \frac{45}{16} - \frac{15}{16} \\ \frac{66}{24} + \frac{39}{24} - \frac{5}{24} \\ \frac{150}{48} + \frac{69}{48} - \frac{55}{48} \end{matrix}] \\ = [\begin{matrix} \frac{33}{4} \\ \frac{25}{6} \\ \frac{41}{12} \end{matrix}] \end{aligned}$

The solutions to the given equation are $x = \frac{33}{4}, y = \frac{25}{6}, z = \frac{41}{12}$

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In Problems 20–25, solve each system of equations using matrices. If the system has no solution, say that it is inconsistent.
$\{\begin{cases} 5 x - 6 y - 3 z = 6 \\ 4 x - 7 y - 2 z = - 3 \\ 3 x + y - 7 z = 1 \end{cases}$

Short Answer

Step by step solution

Step 1. Given information

Step 2. Solve system of equations using matrix

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In Problems 20–25, solve each system of equations using matrices. If the system has no solution, say that it is inconsistent.5x−6y−3z=64x−7y−2z=−33x+y−7z=1

Short Answer

Step by step solution

Step 1. Given information

Step 2. Solve system of equations using matrix

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Applied Mathematics

Geometry

Pure Maths

Mechanics Maths

Calculus

Study anywhere. Anytime. Across all devices.

In Problems 20–25, solve each system of equations using matrices. If the system has no solution, say that it is inconsistent.
$\{\begin{cases} 5 x - 6 y - 3 z = 6 \\ 4 x - 7 y - 2 z = - 3 \\ 3 x + y - 7 z = 1 \end{cases}$