Linear Algebra Examples

$[\begin{array}{c} 1 & 1 & 1 \\ 2 & 2 & 2 \\ 3 & 3 & 3 \end{array}] [\begin{array}{c} x \\ y \\ z \end{array}] = [\begin{array}{c} 3 \\ 6 \\ 9 \end{array}]$

Step 1

Multiply $[\begin{array}{c} 1 & 1 & 1 \\ 2 & 2 & 2 \\ 3 & 3 & 3 \end{array}] [\begin{array}{c} x \\ y \\ z \end{array}]$ .

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Step 1.1

Two matrices can be multiplied if and only if the number of columns in the first matrix is equal to the number of rows in the second matrix. In this case, the first matrix is $3 \times 3$ and the second matrix is $3 \times 1$ .

Step 1.2

Multiply each row in the first matrix by each column in the second matrix.

$[\begin{array}{c} 1 x + 1 y + 1 z \\ 2 x + 2 y + 2 z \\ 3 x + 3 y + 3 z \end{array}] = [\begin{array}{c} 3 \\ 6 \\ 9 \end{array}]$

Step 1.3

Simplify each element of the matrix by multiplying out all the expressions.

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Step 1.3.1

Multiply $x$ by $1$ .

$[\begin{array}{c} x + 1 y + 1 z \\ 2 x + 2 y + 2 z \\ 3 x + 3 y + 3 z \end{array}] = [\begin{array}{c} 3 \\ 6 \\ 9 \end{array}]$

Step 1.3.2

Multiply $y$ by $1$ .

$[\begin{array}{c} x + y + 1 z \\ 2 x + 2 y + 2 z \\ 3 x + 3 y + 3 z \end{array}] = [\begin{array}{c} 3 \\ 6 \\ 9 \end{array}]$

Step 1.3.3

Multiply $z$ by $1$ .

$[\begin{array}{c} x + y + z \\ 2 x + 2 y + 2 z \\ 3 x + 3 y + 3 z \end{array}] = [\begin{array}{c} 3 \\ 6 \\ 9 \end{array}]$

Step 2

Write as a linear system of equations.

$x + y + z = 3$

$2 x + 2 y + 2 z = 6$

$3 x + 3 y + 3 z = 9$

Step 3

Solve the system of equations.

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Step 3.1

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

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Step 3.1.1

Subtract $y$ from both sides of the equation.

$x + z = 3 - y$

$2 x + 2 y + 2 z = 6$

$3 x + 3 y + 3 z = 9$

Step 3.1.2

Subtract $z$ from both sides of the equation.

$x = 3 - y - z$

$2 x + 2 y + 2 z = 6$

$3 x + 3 y + 3 z = 9$

$x = 3 - y - z$

$2 x + 2 y + 2 z = 6$

$3 x + 3 y + 3 z = 9$

Step 3.2

Replace all occurrences of $x$ with $3 - y - z$ in each equation.

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Step 3.2.1

Replace all occurrences of $x$ in $2 x + 2 y + 2 z = 6$ with $3 - y - z$ .

$2 (3 - y - z) + 2 y + 2 z = 6$

$x = 3 - y - z$

$3 x + 3 y + 3 z = 9$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Simplify $2 (3 - y - z) + 2 y + 2 z$ .

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Step 3.2.2.1.1

Simplify each term.

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Step 3.2.2.1.1.1

Apply the distributive property.

$2 \cdot 3 + 2 (- y) + 2 (- z) + 2 y + 2 z = 6$

$x = 3 - y - z$

$3 x + 3 y + 3 z = 9$

Step 3.2.2.1.1.2

Simplify.

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Step 3.2.2.1.1.2.1

Multiply $2$ by $3$ .

$6 + 2 (- y) + 2 (- z) + 2 y + 2 z = 6$

$x = 3 - y - z$

$3 x + 3 y + 3 z = 9$

Step 3.2.2.1.1.2.2

Multiply $- 1$ by $2$ .

$6 - 2 y + 2 (- z) + 2 y + 2 z = 6$

$x = 3 - y - z$

$3 x + 3 y + 3 z = 9$

Step 3.2.2.1.1.2.3

Multiply $- 1$ by $2$ .

$6 - 2 y - 2 z + 2 y + 2 z = 6$

$x = 3 - y - z$

$3 x + 3 y + 3 z = 9$

$6 - 2 y - 2 z + 2 y + 2 z = 6$

$x = 3 - y - z$

$3 x + 3 y + 3 z = 9$

$6 - 2 y - 2 z + 2 y + 2 z = 6$

$x = 3 - y - z$

$3 x + 3 y + 3 z = 9$

Step 3.2.2.1.2

Combine the opposite terms in $6 - 2 y - 2 z + 2 y + 2 z$ .

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Step 3.2.2.1.2.1

Add $- 2 y$ and $2 y$ .

$6 - 2 z + 0 + 2 z = 6$

$x = 3 - y - z$

$3 x + 3 y + 3 z = 9$

Step 3.2.2.1.2.2

Add $6 - 2 z$ and $0$ .

$6 - 2 z + 2 z = 6$

$x = 3 - y - z$

$3 x + 3 y + 3 z = 9$

Step 3.2.2.1.2.3

Add $- 2 z$ and $2 z$ .

$6 + 0 = 6$

$x = 3 - y - z$

$3 x + 3 y + 3 z = 9$

Step 3.2.2.1.2.4

Add $6$ and $0$ .

$6 = 6$

$x = 3 - y - z$

$3 x + 3 y + 3 z = 9$

$6 = 6$

$x = 3 - y - z$

$3 x + 3 y + 3 z = 9$

$6 = 6$

$x = 3 - y - z$

$3 x + 3 y + 3 z = 9$

$6 = 6$

$x = 3 - y - z$

$3 x + 3 y + 3 z = 9$

Step 3.2.3

Replace all occurrences of $x$ in $3 x + 3 y + 3 z = 9$ with $3 - y - z$ .

$3 (3 - y - z) + 3 y + 3 z = 9$

$6 = 6$

$x = 3 - y - z$

Step 3.2.4

Simplify the left side.

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Step 3.2.4.1

Simplify $3 (3 - y - z) + 3 y + 3 z$ .

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Step 3.2.4.1.1

Simplify each term.

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Step 3.2.4.1.1.1

Apply the distributive property.

$3 \cdot 3 + 3 (- y) + 3 (- z) + 3 y + 3 z = 9$

$6 = 6$

$x = 3 - y - z$

Step 3.2.4.1.1.2

Simplify.

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Step 3.2.4.1.1.2.1

Multiply $3$ by $3$ .

$9 + 3 (- y) + 3 (- z) + 3 y + 3 z = 9$

$6 = 6$

$x = 3 - y - z$

Step 3.2.4.1.1.2.2

Multiply $- 1$ by $3$ .

$9 - 3 y + 3 (- z) + 3 y + 3 z = 9$

$6 = 6$

$x = 3 - y - z$

Step 3.2.4.1.1.2.3

Multiply $- 1$ by $3$ .

$9 - 3 y - 3 z + 3 y + 3 z = 9$

$6 = 6$

$x = 3 - y - z$

$9 - 3 y - 3 z + 3 y + 3 z = 9$

$6 = 6$

$x = 3 - y - z$

$9 - 3 y - 3 z + 3 y + 3 z = 9$

$6 = 6$

$x = 3 - y - z$

Step 3.2.4.1.2

Combine the opposite terms in $9 - 3 y - 3 z + 3 y + 3 z$ .

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Step 3.2.4.1.2.1

Add $- 3 y$ and $3 y$ .

$9 - 3 z + 0 + 3 z = 9$

$6 = 6$

$x = 3 - y - z$

Step 3.2.4.1.2.2

Add $9 - 3 z$ and $0$ .

$9 - 3 z + 3 z = 9$

$6 = 6$

$x = 3 - y - z$

Step 3.2.4.1.2.3

Add $- 3 z$ and $3 z$ .

$9 + 0 = 9$

$6 = 6$

$x = 3 - y - z$

Step 3.2.4.1.2.4

Add $9$ and $0$ .

$9 = 9$

$6 = 6$

$x = 3 - y - z$

$9 = 9$

$6 = 6$

$x = 3 - y - z$

$9 = 9$

$6 = 6$

$x = 3 - y - z$

$9 = 9$

$6 = 6$

$x = 3 - y - z$

$9 = 9$

$6 = 6$

$x = 3 - y - z$

Step 3.3

Remove any equations from the system that are always true.

$x = 3 - y - z$