Linear Algebra Examples

$\frac{x + 2}{6} - \frac{y + 6}{3} + \frac{z}{2} = 0$ , $\frac{x + 1}{2} + \frac{y - 1}{2} - \frac{z}{4} = 6$ , $\frac{x - 5}{4} + \frac{y + 1}{3} + \frac{z - 2}{2} = \frac{83}{12}$

Step 1

Simplify.

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Step 1.1

To write $- \frac{y + 6}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{x + 2}{6} - \frac{y + 6}{3} \cdot \frac{2}{2} + \frac{z}{2} = 0, \frac{x + 1}{2} + \frac{y - 1}{2} - \frac{z}{4} = 6, \frac{x - 5}{4} + \frac{y + 1}{3} + \frac{z - 2}{2} = \frac{83}{12}$

Step 1.2

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.2.1

Multiply $\frac{y + 6}{3}$ by $\frac{2}{2}$ .

$\frac{x + 2}{6} - \frac{(y + 6) \cdot 2}{3 \cdot 2} + \frac{z}{2} = 0, \frac{x + 1}{2} + \frac{y - 1}{2} - \frac{z}{4} = 6, \frac{x - 5}{4} + \frac{y + 1}{3} + \frac{z - 2}{2} = \frac{83}{12}$

Step 1.2.2

Multiply $3$ by $2$ .

$\frac{x + 2}{6} - \frac{(y + 6) \cdot 2}{6} + \frac{z}{2} = 0, \frac{x + 1}{2} + \frac{y - 1}{2} - \frac{z}{4} = 6, \frac{x - 5}{4} + \frac{y + 1}{3} + \frac{z - 2}{2} = \frac{83}{12}$

Step 1.3

Combine the numerators over the common denominator.

$\frac{x + 2 - (y + 6) \cdot 2}{6} + \frac{z}{2} = 0, \frac{x + 1}{2} + \frac{y - 1}{2} - \frac{z}{4} = 6, \frac{x - 5}{4} + \frac{y + 1}{3} + \frac{z - 2}{2} = \frac{83}{12}$

Step 1.4

Simplify the numerator.

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Step 1.4.1

Apply the distributive property.

$\frac{x + 2 + (- y - 1 \cdot 6) \cdot 2}{6} + \frac{z}{2} = 0, \frac{x + 1}{2} + \frac{y - 1}{2} - \frac{z}{4} = 6, \frac{x - 5}{4} + \frac{y + 1}{3} + \frac{z - 2}{2} = \frac{83}{12}$

Step 1.4.2

Multiply $- 1$ by $6$ .

$\frac{x + 2 + (- y - 6) \cdot 2}{6} + \frac{z}{2} = 0, \frac{x + 1}{2} + \frac{y - 1}{2} - \frac{z}{4} = 6, \frac{x - 5}{4} + \frac{y + 1}{3} + \frac{z - 2}{2} = \frac{83}{12}$

Step 1.4.3

Apply the distributive property.

$\frac{x + 2 - y \cdot 2 - 6 \cdot 2}{6} + \frac{z}{2} = 0, \frac{x + 1}{2} + \frac{y - 1}{2} - \frac{z}{4} = 6, \frac{x - 5}{4} + \frac{y + 1}{3} + \frac{z - 2}{2} = \frac{83}{12}$

Step 1.4.4

Multiply $2$ by $- 1$ .

$\frac{x + 2 - 2 y - 6 \cdot 2}{6} + \frac{z}{2} = 0, \frac{x + 1}{2} + \frac{y - 1}{2} - \frac{z}{4} = 6, \frac{x - 5}{4} + \frac{y + 1}{3} + \frac{z - 2}{2} = \frac{83}{12}$

Step 1.4.5

Multiply $- 6$ by $2$ .

$\frac{x + 2 - 2 y - 12}{6} + \frac{z}{2} = 0, \frac{x + 1}{2} + \frac{y - 1}{2} - \frac{z}{4} = 6, \frac{x - 5}{4} + \frac{y + 1}{3} + \frac{z - 2}{2} = \frac{83}{12}$

Step 1.4.6

Subtract $12$ from $2$ .

$\frac{x - 2 y - 10}{6} + \frac{z}{2} = 0, \frac{x + 1}{2} + \frac{y - 1}{2} - \frac{z}{4} = 6, \frac{x - 5}{4} + \frac{y + 1}{3} + \frac{z - 2}{2} = \frac{83}{12}$

Step 1.5

To write $\frac{z}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{x - 2 y - 10}{6} + \frac{z}{2} \cdot \frac{3}{3} = 0, \frac{x + 1}{2} + \frac{y - 1}{2} - \frac{z}{4} = 6, \frac{x - 5}{4} + \frac{y + 1}{3} + \frac{z - 2}{2} = \frac{83}{12}$

Step 1.6

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.6.1

Multiply $\frac{z}{2}$ by $\frac{3}{3}$ .

$\frac{x - 2 y - 10}{6} + \frac{z \cdot 3}{2 \cdot 3} = 0, \frac{x + 1}{2} + \frac{y - 1}{2} - \frac{z}{4} = 6, \frac{x - 5}{4} + \frac{y + 1}{3} + \frac{z - 2}{2} = \frac{83}{12}$

Step 1.6.2

Multiply $2$ by $3$ .

$\frac{x - 2 y - 10}{6} + \frac{z \cdot 3}{6} = 0, \frac{x + 1}{2} + \frac{y - 1}{2} - \frac{z}{4} = 6, \frac{x - 5}{4} + \frac{y + 1}{3} + \frac{z - 2}{2} = \frac{83}{12}$

Step 1.7

Combine the numerators over the common denominator.

$\frac{x - 2 y - 10 + z \cdot 3}{6} = 0, \frac{x + 1}{2} + \frac{y - 1}{2} - \frac{z}{4} = 6, \frac{x - 5}{4} + \frac{y + 1}{3} + \frac{z - 2}{2} = \frac{83}{12}$

Step 1.8

Move $3$ to the left of $z$ .

$\frac{x - 2 y - 10 + 3 z}{6} = 0, \frac{x + 1}{2} + \frac{y - 1}{2} - \frac{z}{4} = 6, \frac{x - 5}{4} + \frac{y + 1}{3} + \frac{z - 2}{2} = \frac{83}{12}$

Step 2

Simplify.

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Step 2.1

Combine the numerators over the common denominator.

$\frac{x - 2 y - 10 + 3 z}{6} = 0, \frac{x + 1 + y - 1}{2} + \frac{- z}{4} = 6, \frac{x - 5}{4} + \frac{y + 1}{3} + \frac{z - 2}{2} = \frac{83}{12}$

Step 2.2

Combine the opposite terms in $x + 1 + y - 1$ .

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Step 2.2.1

Subtract $1$ from $1$ .

$\frac{x - 2 y - 10 + 3 z}{6} = 0, \frac{x + y + 0}{2} + \frac{- z}{4} = 6, \frac{x - 5}{4} + \frac{y + 1}{3} + \frac{z - 2}{2} = \frac{83}{12}$

Step 2.2.2

Add $x + y$ and $0$ .

$\frac{x - 2 y - 10 + 3 z}{6} = 0, \frac{x + y}{2} + \frac{- z}{4} = 6, \frac{x - 5}{4} + \frac{y + 1}{3} + \frac{z - 2}{2} = \frac{83}{12}$

Step 2.3

Move the negative in front of the fraction.

$\frac{x - 2 y - 10 + 3 z}{6} = 0, \frac{x + y}{2} - \frac{z}{4} = 6, \frac{x - 5}{4} + \frac{y + 1}{3} + \frac{z - 2}{2} = \frac{83}{12}$

Step 2.4

To write $\frac{x + y}{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{x - 2 y - 10 + 3 z}{6} = 0, \frac{x + y}{2} \cdot \frac{2}{2} - \frac{z}{4} = 6, \frac{x - 5}{4} + \frac{y + 1}{3} + \frac{z - 2}{2} = \frac{83}{12}$

Step 2.5

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.5.1

Multiply $\frac{x + y}{2}$ by $\frac{2}{2}$ .

$\frac{x - 2 y - 10 + 3 z}{6} = 0, \frac{(x + y) \cdot 2}{2 \cdot 2} - \frac{z}{4} = 6, \frac{x - 5}{4} + \frac{y + 1}{3} + \frac{z - 2}{2} = \frac{83}{12}$

Step 2.5.2

Multiply $2$ by $2$ .

$\frac{x - 2 y - 10 + 3 z}{6} = 0, \frac{(x + y) \cdot 2}{4} - \frac{z}{4} = 6, \frac{x - 5}{4} + \frac{y + 1}{3} + \frac{z - 2}{2} = \frac{83}{12}$

Step 2.6

Combine the numerators over the common denominator.

$\frac{x - 2 y - 10 + 3 z}{6} = 0, \frac{(x + y) \cdot 2 - z}{4} = 6, \frac{x - 5}{4} + \frac{y + 1}{3} + \frac{z - 2}{2} = \frac{83}{12}$

Step 2.7

Simplify the numerator.

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Step 2.7.1

Apply the distributive property.

$\frac{x - 2 y - 10 + 3 z}{6} = 0, \frac{x \cdot 2 + y \cdot 2 - z}{4} = 6, \frac{x - 5}{4} + \frac{y + 1}{3} + \frac{z - 2}{2} = \frac{83}{12}$

Step 2.7.2

Move $2$ to the left of $x$ .

$\frac{x - 2 y - 10 + 3 z}{6} = 0, \frac{2 \cdot x + y \cdot 2 - z}{4} = 6, \frac{x - 5}{4} + \frac{y + 1}{3} + \frac{z - 2}{2} = \frac{83}{12}$

Step 2.7.3

Move $2$ to the left of $y$ .

$\frac{x - 2 y - 10 + 3 z}{6} = 0, \frac{2 \cdot x + 2 \cdot y - z}{4} = 6, \frac{x - 5}{4} + \frac{y + 1}{3} + \frac{z - 2}{2} = \frac{83}{12}$

Step 2.7.4

Multiply $2$ by $y$ .

$\frac{x - 2 y - 10 + 3 z}{6} = 0, \frac{2 x + 2 y - z}{4} = 6, \frac{x - 5}{4} + \frac{y + 1}{3} + \frac{z - 2}{2} = \frac{83}{12}$

Step 3

Simplify.

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Step 3.1

To write $\frac{x - 5}{4}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{x - 2 y - 10 + 3 z}{6} = 0, \frac{2 x + 2 y - z}{4} = 6, \frac{x - 5}{4} \cdot \frac{3}{3} + \frac{y + 1}{3} + \frac{z - 2}{2} = \frac{83}{12}$

Step 3.2

To write $\frac{y + 1}{3}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{x - 2 y - 10 + 3 z}{6} = 0, \frac{2 x + 2 y - z}{4} = 6, \frac{x - 5}{4} \cdot \frac{3}{3} + \frac{y + 1}{3} \cdot \frac{4}{4} + \frac{z - 2}{2} = \frac{83}{12}$

Step 3.3

Write each expression with a common denominator of $12$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.3.1

Multiply $\frac{x - 5}{4}$ by $\frac{3}{3}$ .

$\frac{x - 2 y - 10 + 3 z}{6} = 0, \frac{2 x + 2 y - z}{4} = 6, \frac{(x - 5) \cdot 3}{4 \cdot 3} + \frac{y + 1}{3} \cdot \frac{4}{4} + \frac{z - 2}{2} = \frac{83}{12}$

Step 3.3.2

Multiply $4$ by $3$ .

$\frac{x - 2 y - 10 + 3 z}{6} = 0, \frac{2 x + 2 y - z}{4} = 6, \frac{(x - 5) \cdot 3}{12} + \frac{y + 1}{3} \cdot \frac{4}{4} + \frac{z - 2}{2} = \frac{83}{12}$

Step 3.3.3

Multiply $\frac{y + 1}{3}$ by $\frac{4}{4}$ .

$\frac{x - 2 y - 10 + 3 z}{6} = 0, \frac{2 x + 2 y - z}{4} = 6, \frac{(x - 5) \cdot 3}{12} + \frac{(y + 1) \cdot 4}{3 \cdot 4} + \frac{z - 2}{2} = \frac{83}{12}$

Step 3.3.4

Multiply $3$ by $4$ .

$\frac{x - 2 y - 10 + 3 z}{6} = 0, \frac{2 x + 2 y - z}{4} = 6, \frac{(x - 5) \cdot 3}{12} + \frac{(y + 1) \cdot 4}{12} + \frac{z - 2}{2} = \frac{83}{12}$

Step 3.4

Combine the numerators over the common denominator.

$\frac{x - 2 y - 10 + 3 z}{6} = 0, \frac{2 x + 2 y - z}{4} = 6, \frac{(x - 5) \cdot 3 + (y + 1) \cdot 4}{12} + \frac{z - 2}{2} = \frac{83}{12}$

Step 3.5

Simplify the numerator.

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Step 3.5.1

Apply the distributive property.

$\frac{x - 2 y - 10 + 3 z}{6} = 0, \frac{2 x + 2 y - z}{4} = 6, \frac{x \cdot 3 - 5 \cdot 3 + (y + 1) \cdot 4}{12} + \frac{z - 2}{2} = \frac{83}{12}$

Step 3.5.2

Move $3$ to the left of $x$ .

$\frac{x - 2 y - 10 + 3 z}{6} = 0, \frac{2 x + 2 y - z}{4} = 6, \frac{3 \cdot x - 5 \cdot 3 + (y + 1) \cdot 4}{12} + \frac{z - 2}{2} = \frac{83}{12}$

Step 3.5.3

Multiply $- 5$ by $3$ .

$\frac{x - 2 y - 10 + 3 z}{6} = 0, \frac{2 x + 2 y - z}{4} = 6, \frac{3 \cdot x - 15 + (y + 1) \cdot 4}{12} + \frac{z - 2}{2} = \frac{83}{12}$

Step 3.5.4

Apply the distributive property.

$\frac{x - 2 y - 10 + 3 z}{6} = 0, \frac{2 x + 2 y - z}{4} = 6, \frac{3 x - 15 + y \cdot 4 + 1 \cdot 4}{12} + \frac{z - 2}{2} = \frac{83}{12}$

Step 3.5.5

Move $4$ to the left of $y$ .

$\frac{x - 2 y - 10 + 3 z}{6} = 0, \frac{2 x + 2 y - z}{4} = 6, \frac{3 x - 15 + 4 \cdot y + 1 \cdot 4}{12} + \frac{z - 2}{2} = \frac{83}{12}$

Step 3.5.6

Multiply $4$ by $1$ .

$\frac{x - 2 y - 10 + 3 z}{6} = 0, \frac{2 x + 2 y - z}{4} = 6, \frac{3 x - 15 + 4 \cdot y + 4}{12} + \frac{z - 2}{2} = \frac{83}{12}$

Step 3.5.7

Add $- 15$ and $4$ .

$\frac{x - 2 y - 10 + 3 z}{6} = 0, \frac{2 x + 2 y - z}{4} = 6, \frac{3 x + 4 y - 11}{12} + \frac{z - 2}{2} = \frac{83}{12}$

Step 3.6

To write $\frac{z - 2}{2}$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$\frac{x - 2 y - 10 + 3 z}{6} = 0, \frac{2 x + 2 y - z}{4} = 6, \frac{3 x + 4 y - 11}{12} + \frac{z - 2}{2} \cdot \frac{6}{6} = \frac{83}{12}$

Step 3.7

Write each expression with a common denominator of $12$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.7.1

Multiply $\frac{z - 2}{2}$ by $\frac{6}{6}$ .

$\frac{x - 2 y - 10 + 3 z}{6} = 0, \frac{2 x + 2 y - z}{4} = 6, \frac{3 x + 4 y - 11}{12} + \frac{(z - 2) \cdot 6}{2 \cdot 6} = \frac{83}{12}$

Step 3.7.2

Multiply $2$ by $6$ .

$\frac{x - 2 y - 10 + 3 z}{6} = 0, \frac{2 x + 2 y - z}{4} = 6, \frac{3 x + 4 y - 11}{12} + \frac{(z - 2) \cdot 6}{12} = \frac{83}{12}$

Step 3.8

Combine the numerators over the common denominator.

$\frac{x - 2 y - 10 + 3 z}{6} = 0, \frac{2 x + 2 y - z}{4} = 6, \frac{3 x + 4 y - 11 + (z - 2) \cdot 6}{12} = \frac{83}{12}$

Step 3.9

Simplify the numerator.

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Step 3.9.1

Apply the distributive property.

$\frac{x - 2 y - 10 + 3 z}{6} = 0, \frac{2 x + 2 y - z}{4} = 6, \frac{3 x + 4 y - 11 + z \cdot 6 - 2 \cdot 6}{12} = \frac{83}{12}$

Step 3.9.2

Move $6$ to the left of $z$ .

$\frac{x - 2 y - 10 + 3 z}{6} = 0, \frac{2 x + 2 y - z}{4} = 6, \frac{3 x + 4 y - 11 + 6 \cdot z - 2 \cdot 6}{12} = \frac{83}{12}$

Step 3.9.3

Multiply $- 2$ by $6$ .

$\frac{x - 2 y - 10 + 3 z}{6} = 0, \frac{2 x + 2 y - z}{4} = 6, \frac{3 x + 4 y - 11 + 6 \cdot z - 12}{12} = \frac{83}{12}$

Step 3.9.4

Subtract $12$ from $- 11$ .

$\frac{x - 2 y - 10 + 3 z}{6} = 0, \frac{2 x + 2 y - z}{4} = 6, \frac{3 x + 4 y + 6 z - 23}{12} = \frac{83}{12}$

Step 4

Write the system of equations in matrix form.

$[\begin{array}{c} \frac{1}{6} & 0 & 0 & 0 \\ \frac{1}{4} & 0 & 0 & 6 \\ \frac{1}{12} & 0 & 0 & \frac{83}{12} \end{array}]$

Step 5

Find the reduced row echelon form.

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Step 5.1

Multiply each element of $R_{1}$ by $6$ to make the entry at $1, 1$ a $1$ .

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Step 5.1.1

Multiply each element of $R_{1}$ by $6$ to make the entry at $1, 1$ a $1$ .

$[\begin{array}{c} 6 (\frac{1}{6}) & 6 \cdot 0 & 6 \cdot 0 & 6 \cdot 0 \\ \frac{1}{4} & 0 & 0 & 6 \\ \frac{1}{12} & 0 & 0 & \frac{83}{12} \end{array}]$

Step 5.1.2

Simplify $R_{1}$ .

$[\begin{array}{c} 1 & 0 & 0 & 0 \\ \frac{1}{4} & 0 & 0 & 6 \\ \frac{1}{12} & 0 & 0 & \frac{83}{12} \end{array}]$

Step 5.2

Perform the row operation $R_{2} = R_{2} - \frac{1}{4} R_{1}$ to make the entry at $2, 1$ a $0$ .

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Step 5.2.1

Perform the row operation $R_{2} = R_{2} - \frac{1}{4} R_{1}$ to make the entry at $2, 1$ a $0$ .

$[\begin{array}{c} 1 & 0 & 0 & 0 \\ \frac{1}{4} - \frac{1}{4} \cdot 1 & 0 - \frac{1}{4} \cdot 0 & 0 - \frac{1}{4} \cdot 0 & 6 - \frac{1}{4} \cdot 0 \\ \frac{1}{12} & 0 & 0 & \frac{83}{12} \end{array}]$

Step 5.2.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 6 \\ \frac{1}{12} & 0 & 0 & \frac{83}{12} \end{array}]$

Step 5.3

Perform the row operation $R_{3} = R_{3} - \frac{1}{12} R_{1}$ to make the entry at $3, 1$ a $0$ .

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Step 5.3.1

Perform the row operation $R_{3} = R_{3} - \frac{1}{12} R_{1}$ to make the entry at $3, 1$ a $0$ .

$[\begin{array}{c} 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 6 \\ \frac{1}{12} - \frac{1}{12} \cdot 1 & 0 - \frac{1}{12} \cdot 0 & 0 - \frac{1}{12} \cdot 0 & \frac{83}{12} - \frac{1}{12} \cdot 0 \end{array}]$

Step 5.3.2

Simplify $R_{3}$ .

$[\begin{array}{c} 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 6 \\ 0 & 0 & 0 & \frac{83}{12} \end{array}]$

Step 5.4

Multiply each element of $R_{2}$ by $\frac{1}{6}$ to make the entry at $2, 4$ a $1$ .

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Step 5.4.1

Multiply each element of $R_{2}$ by $\frac{1}{6}$ to make the entry at $2, 4$ a $1$ .

$[\begin{array}{c} 1 & 0 & 0 & 0 \\ \frac{0}{6} & \frac{0}{6} & \frac{0}{6} & \frac{6}{6} \\ 0 & 0 & 0 & \frac{83}{12} \end{array}]$

Step 5.4.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & \frac{83}{12} \end{array}]$

Step 5.5

Perform the row operation $R_{3} = R_{3} - \frac{83}{12} R_{2}$ to make the entry at $3, 4$ a $0$ .

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Step 5.5.1

Perform the row operation $R_{3} = R_{3} - \frac{83}{12} R_{2}$ to make the entry at $3, 4$ a $0$ .

$[\begin{array}{c} 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 - \frac{83}{12} \cdot 0 & 0 - \frac{83}{12} \cdot 0 & 0 - \frac{83}{12} \cdot 0 & \frac{83}{12} - \frac{83}{12} \cdot 1 \end{array}]$

Step 5.5.2

Simplify $R_{3}$ .

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

Step 6

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

$x = 0$

$0 = 1$

Step 7

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

No solution