Chapter 1: Q15E (page 19)

Solve the linear system of equations. You may use technology.
$| \begin{matrix} 3 x + 5 y + 3 z = 25 \\ 7 X + 9 y + 19 z = 65 \\ 4 X + 5 y + 11 z = 5 \end{matrix} |$

Short Answer

Expert verified

For the given system of equation there is unique solution. $X = 4, y = 2 . z = 1$ .

Step by step solution

Augmented matrix

Solving system of equations using gauss Jordan’s elimination method.

First of all we will make the augmented matrix to the corresponding system of equations.

$|\begin{matrix} 3 X + 5 y + 3 z = 25 \\ 7 X + 9 y + 19 z = 65 \\ - 4 X + 5 y + 11 z = 5 \end{matrix}|$

Corresponding augmented matrix

$[\begin{matrix} 3 & 5 & 3 \\ 7 & 9 & 19 \\ - 4 & 5 & 11 \end{matrix} \begin{matrix} 25 \\ 65 \\ 5 \end{matrix}]$

Solving the Augmented matrix

$[\begin{matrix} 3 & 5 & 3 \\ 7 & 9 & 19 \\ - 4 & 5 & 11 \end{matrix} \begin{matrix} 25 \\ 65 \\ 5 \end{matrix}]$

$R_{1} \leftrightarrow R_{3} R_{1} \to (- 1) R_{1} - R_{3} [\begin{matrix} 1 & - 10 & - 14 \\ 7 & 9 & 19 \\ 3 & 5 & 3 \end{matrix} \begin{matrix} 30 \\ 65 \\ 25 \end{matrix}]$

$R_{2} \to R_{2} - 7 R_{1} R_{3} \to R_{3} - 3 R_{1} [\begin{matrix} 1 & - 10 & - 14 \\ 0 & 79 & 117 \\ 0 & 35 & 45 \end{matrix} \begin{matrix} - 30 \\ 275 \\ 115 \end{matrix}] R_{2} \to \frac{1}{79} R_{2} R_{3} \to \frac{1}{5} R_{3} [\begin{matrix} 1 & - 10 & - 14 \\ 0 & 1 & \frac{117}{79} \\ 0 & 7 & 9 \end{matrix} \begin{matrix} - 30 \\ 79 \\ 23 \end{matrix}] R_{1} \to R_{1} + 10 R_{2} R_{3} \to R_{3} - 7 R_{2} [\begin{matrix} 1 & 0 & \frac{64}{79} \\ 0 & 1 & \frac{117}{79} \\ 0 & 0 & \frac{- 108}{79} \end{matrix} \begin{matrix} \frac{380}{79} \\ \frac{275}{79} \\ \frac{- 108}{79} \end{matrix}]$

$R_{3} \to \frac{- 79}{108} R_{3} R_{1} \to R_{1} - \frac{64}{79} R_{2} R_{2} \to R_{2} - \frac{275}{79} R_{2} [\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} \begin{matrix} 4 \\ 2 \\ 1 \end{matrix}]$

Step 3: Value of variables

Corresponding system of equations will be

$|\begin{matrix} x = 4 \\ y = 2 \\ z = 1 \end{matrix}|$

Hence, the solution of equation will be $x = 4, y = 2, z = 1$

Final answer

For the given system of equation there is unique solution $x = 4, y = 2, z = 1$ .

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Solve the linear system of equations. You may use technology.
$| \begin{matrix} 3 x + 5 y + 3 z = 25 \\ 7 X + 9 y + 19 z = 65 \\ 4 X + 5 y + 11 z = 5 \end{matrix} |$

Short Answer

Step by step solution

Augmented matrix

Solving the Augmented matrix

Step 3: Value of variables

Final answer

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Solve the linear system of equations. You may use technology.|3x+5y+3z=257X+9y+19z=654X+5y+11z=5|

Short Answer

Step by step solution

Augmented matrix

Solving the Augmented matrix

Step 3: Value of variables

Final answer

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

Decision Maths

Mechanics Maths

Pure Maths

Calculus

Statistics

Study anywhere. Anytime. Across all devices.

Solve the linear system of equations. You may use technology.
$| \begin{matrix} 3 x + 5 y + 3 z = 25 \\ 7 X + 9 y + 19 z = 65 \\ 4 X + 5 y + 11 z = 5 \end{matrix} |$