Linear Algebra Examples

$x + 2 y - 4 z = 4$ , $- 3 x - 6 y + 12 z = - 12$

Step 1

Write the system of equations in matrix form.

$[\begin{array}{c} 1 & 2 & - 4 & 4 \\ - 3 & - 6 & 12 & - 12 \end{array}]$

Step 2

Perform the row operation $R_{2} = R_{2} + 3 R_{1}$ to make the entry at $2, 1$ a $0$ .

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Step 2.1

Perform the row operation $R_{2} = R_{2} + 3 R_{1}$ to make the entry at $2, 1$ a $0$ .

$[\begin{array}{c} 1 & 2 & - 4 & 4 \\ - 3 + 3 \cdot 1 & - 6 + 3 \cdot 2 & 12 + 3 \cdot - 4 & - 12 + 3 \cdot 4 \end{array}]$

Step 2.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & 2 & - 4 & 4 \\ 0 & 0 & 0 & 0 \end{array}]$

Step 3

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

$x + 2 y - 4 z = 4$

Step 4

Solve the equation for $y$ .

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Step 4.1

Move all terms not containing $y$ to the right side of the equation.

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Step 4.1.1

Subtract $x$ from both sides of the equation.

$2 y - 4 z = 4 - x$

Step 4.1.2

Add $4 z$ to both sides of the equation.

$2 y = 4 - x + 4 z$

Step 4.2

Divide each term in $2 y = 4 - x + 4 z$ by $2$ and simplify.

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Step 4.2.1

Divide each term in $2 y = 4 - x + 4 z$ by $2$ .

$\frac{2 y}{2} = \frac{4}{2} + \frac{- x}{2} + \frac{4 z}{2}$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Cancel the common factor of $2$ .

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Step 4.2.2.1.1

Cancel the common factor.

$\frac{2 y}{2} = \frac{4}{2} + \frac{- x}{2} + \frac{4 z}{2}$

Step 4.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{4}{2} + \frac{- x}{2} + \frac{4 z}{2}$

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

Simplify each term.

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Step 4.2.3.1.1

Divide $4$ by $2$ .

$y = 2 + \frac{- x}{2} + \frac{4 z}{2}$

Step 4.2.3.1.2

Move the negative in front of the fraction.

$y = 2 - \frac{x}{2} + \frac{4 z}{2}$

Step 4.2.3.1.3

Cancel the common factor of $4$ and $2$ .

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Step 4.2.3.1.3.1

Factor $2$ out of $4 z$ .

$y = 2 - \frac{x}{2} + \frac{2 (2 z)}{2}$

Step 4.2.3.1.3.2

Cancel the common factors.

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Step 4.2.3.1.3.2.1

Factor $2$ out of $2$ .

$y = 2 - \frac{x}{2} + \frac{2 (2 z)}{2 (1)}$

Step 4.2.3.1.3.2.2

Cancel the common factor.

$y = 2 - \frac{x}{2} + \frac{2 (2 z)}{2 \cdot 1}$

Step 4.2.3.1.3.2.3

Rewrite the expression.

$y = 2 - \frac{x}{2} + \frac{2 z}{1}$

Step 4.2.3.1.3.2.4

Divide $2 z$ by $1$ .

$y = 2 - \frac{x}{2} + 2 z$

Step 5

The solution is the set of ordered pairs that makes the system true.

$(x, 2 - \frac{x}{2} + 2 z, z)$

Step 6

Decompose a solution vector by re-arranging each equation represented in the row-reduced form of the augmented matrix by solving for the dependent variable in each row yields the vector equality.

$X = [\begin{array}{c} x \\ y \\ z \end{array}] = [\begin{array}{c} x \\ 2 - \frac{x}{2} + 2 z \\ z \end{array}]$