Linear Algebra Examples

9 x + 9 x - 7 x = 6

$9 x + 9 x - 7 x = 6$ ,

- 7 x + 0 - 1 x = - 10

$- 7 x + 0 - 1 x = - 10$ ,

9 x + 6 x + 8 x = 45

$9 x + 6 x + 8 x = 45$

Step 1

Simplify.

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Step 1.1

Add

9 x

$9 x$ and

9 x

$9 x$ .

18 x - 7 x = 6, - 7 x + 0 - 1 x = - 10, 9 x + 6 x + 8 x = 45

$18 x - 7 x = 6, - 7 x + 0 - 1 x = - 10, 9 x + 6 x + 8 x = 45$

Step 1.2

Subtract

7 x

$7 x$ from

18 x

$18 x$ .

11 x = 6, - 7 x + 0 - 1 x = - 10, 9 x + 6 x + 8 x = 45

$11 x = 6, - 7 x + 0 - 1 x = - 10, 9 x + 6 x + 8 x = 45$

11 x = 6, - 7 x + 0 - 1 x = - 10, 9 x + 6 x + 8 x = 45

$11 x = 6, - 7 x + 0 - 1 x = - 10, 9 x + 6 x + 8 x = 45$

Step 2

Simplify.

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Step 2.1

Add

- 7 x

$- 7 x$ and

0

$0$ .

11 x = 6, - 7 x - 1 x = - 10, 9 x + 6 x + 8 x = 45

$11 x = 6, - 7 x - 1 x = - 10, 9 x + 6 x + 8 x = 45$

Step 2.2

Rewrite

- 1 x

$- 1 x$ as

- x

$- x$ .

11 x = 6, - 7 x - x = - 10, 9 x + 6 x + 8 x = 45

$11 x = 6, - 7 x - x = - 10, 9 x + 6 x + 8 x = 45$

Step 2.3

Subtract

x

$x$ from

- 7 x

$- 7 x$ .

11 x = 6, - 8 x = - 10, 9 x + 6 x + 8 x = 45

$11 x = 6, - 8 x = - 10, 9 x + 6 x + 8 x = 45$

11 x = 6, - 8 x = - 10, 9 x + 6 x + 8 x = 45

$11 x = 6, - 8 x = - 10, 9 x + 6 x + 8 x = 45$

Step 3

Simplify.

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Step 3.1

Add

9 x

$9 x$ and

6 x

$6 x$ .

11 x = 6, - 8 x = - 10, 15 x + 8 x = 45

$11 x = 6, - 8 x = - 10, 15 x + 8 x = 45$

Step 3.2

Add

15 x

$15 x$ and

8 x

$8 x$ .

11 x = 6, - 8 x = - 10, 23 x = 45

$11 x = 6, - 8 x = - 10, 23 x = 45$

11 x = 6, - 8 x = - 10, 23 x = 45

$11 x = 6, - 8 x = - 10, 23 x = 45$

Step 4

Write the system of equations in matrix form.

[\begin{array}{c} 11 & 6 \\ - 8 & - 10 \\ 23 & 45 \end{array}]

$[\begin{array}{c} 11 & 6 \\ - 8 & - 10 \\ 23 & 45 \end{array}]$

Step 5

Find the reduced row echelon form.

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Step 5.1

Multiply each element of

R_{1}

$R_{1}$ by

\frac{1}{11}

$\frac{1}{11}$ to make the entry at

1, 1

$1, 1$ a

1

$1$ .

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Step 5.1.1

Multiply each element of

R_{1}

$R_{1}$ by

\frac{1}{11}

$\frac{1}{11}$ to make the entry at

1, 1

$1, 1$ a

1

$1$ .

[\begin{array}{c} \frac{11}{11} & \frac{6}{11} \\ - 8 & - 10 \\ 23 & 45 \end{array}]

$[\begin{array}{c} \frac{11}{11} & \frac{6}{11} \\ - 8 & - 10 \\ 23 & 45 \end{array}]$

Step 5.1.2

Simplify

R_{1}

$R_{1}$ .

[\begin{array}{c} 1 & \frac{6}{11} \\ - 8 & - 10 \\ 23 & 45 \end{array}]

$[\begin{array}{c} 1 & \frac{6}{11} \\ - 8 & - 10 \\ 23 & 45 \end{array}]$

[\begin{array}{c} 1 & \frac{6}{11} \\ - 8 & - 10 \\ 23 & 45 \end{array}]

$[\begin{array}{c} 1 & \frac{6}{11} \\ - 8 & - 10 \\ 23 & 45 \end{array}]$

Step 5.2

Perform the row operation

R_{2} = R_{2} + 8 R_{1}

$R_{2} = R_{2} + 8 R_{1}$ to make the entry at

2, 1

$2, 1$ a

0

$0$ .

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Step 5.2.1

Perform the row operation

R_{2} = R_{2} + 8 R_{1}

$R_{2} = R_{2} + 8 R_{1}$ to make the entry at

2, 1

$2, 1$ a

0

$0$ .

[\begin{array}{c} 1 & \frac{6}{11} \\ - 8 + 8 \cdot 1 & - 10 + 8 (\frac{6}{11}) \\ 23 & 45 \end{array}]

$[\begin{array}{c} 1 & \frac{6}{11} \\ - 8 + 8 \cdot 1 & - 10 + 8 (\frac{6}{11}) \\ 23 & 45 \end{array}]$

Step 5.2.2

Simplify

R_{2}

$R_{2}$ .

[\begin{array}{c} 1 & \frac{6}{11} \\ 0 & - \frac{62}{11} \\ 23 & 45 \end{array}]

$[\begin{array}{c} 1 & \frac{6}{11} \\ 0 & - \frac{62}{11} \\ 23 & 45 \end{array}]$

[\begin{array}{c} 1 & \frac{6}{11} \\ 0 & - \frac{62}{11} \\ 23 & 45 \end{array}]

$[\begin{array}{c} 1 & \frac{6}{11} \\ 0 & - \frac{62}{11} \\ 23 & 45 \end{array}]$

Step 5.3

Perform the row operation

R_{3} = R_{3} - 23 R_{1}

$R_{3} = R_{3} - 23 R_{1}$ to make the entry at

3, 1

$3, 1$ a

0

$0$ .

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Step 5.3.1

Perform the row operation

R_{3} = R_{3} - 23 R_{1}

$R_{3} = R_{3} - 23 R_{1}$ to make the entry at

3, 1

$3, 1$ a

0

$0$ .

[\begin{array}{c} 1 & \frac{6}{11} \\ 0 & - \frac{62}{11} \\ 23 - 23 \cdot 1 & 45 - 23 (\frac{6}{11}) \end{array}]

$[\begin{array}{c} 1 & \frac{6}{11} \\ 0 & - \frac{62}{11} \\ 23 - 23 \cdot 1 & 45 - 23 (\frac{6}{11}) \end{array}]$

Step 5.3.2

Simplify

R_{3}

$R_{3}$ .

[\begin{array}{c} 1 & \frac{6}{11} \\ 0 & - \frac{62}{11} \\ 0 & \frac{357}{11} \end{array}]

$[\begin{array}{c} 1 & \frac{6}{11} \\ 0 & - \frac{62}{11} \\ 0 & \frac{357}{11} \end{array}]$

[\begin{array}{c} 1 & \frac{6}{11} \\ 0 & - \frac{62}{11} \\ 0 & \frac{357}{11} \end{array}]

$[\begin{array}{c} 1 & \frac{6}{11} \\ 0 & - \frac{62}{11} \\ 0 & \frac{357}{11} \end{array}]$

Step 5.4

Multiply each element of

R_{2}

$R_{2}$ by

- \frac{11}{62}

$- \frac{11}{62}$ to make the entry at

2, 2

$2, 2$ a

1

$1$ .

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Step 5.4.1

Multiply each element of

R_{2}

$R_{2}$ by

- \frac{11}{62}

$- \frac{11}{62}$ to make the entry at

2, 2

$2, 2$ a

1

$1$ .

[\begin{array}{c} 1 & \frac{6}{11} \\ - \frac{11}{62} \cdot 0 & - \frac{11}{62} (- \frac{62}{11}) \\ 0 & \frac{357}{11} \end{array}]

$[\begin{array}{c} 1 & \frac{6}{11} \\ - \frac{11}{62} \cdot 0 & - \frac{11}{62} (- \frac{62}{11}) \\ 0 & \frac{357}{11} \end{array}]$

Step 5.4.2

Simplify

R_{2}

$R_{2}$ .

[\begin{array}{c} 1 & \frac{6}{11} \\ 0 & 1 \\ 0 & \frac{357}{11} \end{array}]

$[\begin{array}{c} 1 & \frac{6}{11} \\ 0 & 1 \\ 0 & \frac{357}{11} \end{array}]$

[\begin{array}{c} 1 & \frac{6}{11} \\ 0 & 1 \\ 0 & \frac{357}{11} \end{array}]

$[\begin{array}{c} 1 & \frac{6}{11} \\ 0 & 1 \\ 0 & \frac{357}{11} \end{array}]$

Step 5.5

Perform the row operation

R_{3} = R_{3} - \frac{357}{11} R_{2}

$R_{3} = R_{3} - \frac{357}{11} R_{2}$ to make the entry at

3, 2

$3, 2$ a

0

$0$ .

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Step 5.5.1

Perform the row operation

R_{3} = R_{3} - \frac{357}{11} R_{2}

$R_{3} = R_{3} - \frac{357}{11} R_{2}$ to make the entry at

3, 2

$3, 2$ a

0

$0$ .

[\begin{array}{c} 1 & \frac{6}{11} \\ 0 & 1 \\ 0 - \frac{357}{11} \cdot 0 & \frac{357}{11} - \frac{357}{11} \cdot 1 \end{array}]

$[\begin{array}{c} 1 & \frac{6}{11} \\ 0 & 1 \\ 0 - \frac{357}{11} \cdot 0 & \frac{357}{11} - \frac{357}{11} \cdot 1 \end{array}]$

Step 5.5.2

Simplify

R_{3}

$R_{3}$ .

[\begin{array}{c} 1 & \frac{6}{11} \\ 0 & 1 \\ 0 & 0 \end{array}]

$[\begin{array}{c} 1 & \frac{6}{11} \\ 0 & 1 \\ 0 & 0 \end{array}]$

[\begin{array}{c} 1 & \frac{6}{11} \\ 0 & 1 \\ 0 & 0 \end{array}]

$[\begin{array}{c} 1 & \frac{6}{11} \\ 0 & 1 \\ 0 & 0 \end{array}]$

Step 5.6

Perform the row operation

R_{1} = R_{1} - \frac{6}{11} R_{2}

$R_{1} = R_{1} - \frac{6}{11} R_{2}$ to make the entry at

1, 2

$1, 2$ a

0

$0$ .

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Step 5.6.1

Perform the row operation

R_{1} = R_{1} - \frac{6}{11} R_{2}

$R_{1} = R_{1} - \frac{6}{11} R_{2}$ to make the entry at

1, 2

$1, 2$ a

0

$0$ .

[\begin{array}{c} 1 - \frac{6}{11} \cdot 0 & \frac{6}{11} - \frac{6}{11} \cdot 1 \\ 0 & 1 \\ 0 & 0 \end{array}]

$[\begin{array}{c} 1 - \frac{6}{11} \cdot 0 & \frac{6}{11} - \frac{6}{11} \cdot 1 \\ 0 & 1 \\ 0 & 0 \end{array}]$

Step 5.6.2

Simplify

R_{1}

$R_{1}$ .

[\begin{array}{c} 1 & 0 \\ 0 & 1 \\ 0 & 0 \end{array}]

$[\begin{array}{c} 1 & 0 \\ 0 & 1 \\ 0 & 0 \end{array}]$

[\begin{array}{c} 1 & 0 \\ 0 & 1 \\ 0 & 0 \end{array}]

$[\begin{array}{c} 1 & 0 \\ 0 & 1 \\ 0 & 0 \end{array}]$

[\begin{array}{c} 1 & 0 \\ 0 & 1 \\ 0 & 0 \end{array}]

$[\begin{array}{c} 1 & 0 \\ 0 & 1 \\ 0 & 0 \end{array}]$

Step 6

Use the result matrix to declare the final solutions to the system of equations.

x = 0

$x = 0$

0 = 1

$0 = 1$

Step 7

Since

0 \neq 1

$0 \neq 1$ , there are no solutions.

No solution