Linear Algebra Examples

$2 x + [\begin{array}{c} 4 & - 5 \\ 1 & - 12 \end{array}] = [\begin{array}{c} 10 & 1 \\ - 7 & - 2 \end{array}]$

Step 1

Move all terms not containing a variable to the right side.

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Step 1.1

Subtract $[\begin{array}{c} 4 & - 5 \\ 1 & - 12 \end{array}]$ from both sides of the equation.

$2 x = [\begin{array}{c} 10 & 1 \\ - 7 & - 2 \end{array}] - [\begin{array}{c} 4 & - 5 \\ 1 & - 12 \end{array}]$

Step 2

Simplify the right side.

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Step 2.1

Subtract the corresponding elements.

$2 x = [\begin{array}{c} 10 - 4 & 1 + 5 \\ - 7 - 1 & - 2 + 12 \end{array}]$

Step 2.2

Simplify each element.

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Step 2.2.1

Subtract $4$ from $10$ .

$2 x = [\begin{array}{c} 6 & 1 + 5 \\ - 7 - 1 & - 2 + 12 \end{array}]$

Step 2.2.2

Add $1$ and $5$ .

$2 x = [\begin{array}{c} 6 & 6 \\ - 7 - 1 & - 2 + 12 \end{array}]$

Step 2.2.3

Subtract $1$ from $- 7$ .

$2 x = [\begin{array}{c} 6 & 6 \\ - 8 & - 2 + 12 \end{array}]$

Step 2.2.4

Add $- 2$ and $12$ .

$2 x = [\begin{array}{c} 6 & 6 \\ - 8 & 10 \end{array}]$

Step 3

Multiply both sides by $\frac{1}{2}$ .

$\frac{1}{2} (2 x) = \frac{1}{2} [\begin{array}{c} 6 & 6 \\ - 8 & 10 \end{array}]$

Step 4

Simplify.

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Step 4.1

Cancel the common factor of $2$ .

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Step 4.1.1

Factor $2$ out of $2 x$ .

$\frac{1}{2} (2 (x)) = \frac{1}{2} [\begin{array}{c} 6 & 6 \\ - 8 & 10 \end{array}]$

Step 4.1.2

Cancel the common factor.

$\frac{1}{2} (2 x) = \frac{1}{2} [\begin{array}{c} 6 & 6 \\ - 8 & 10 \end{array}]$

Step 4.1.3

Rewrite the expression.

$x = \frac{1}{2} [\begin{array}{c} 6 & 6 \\ - 8 & 10 \end{array}]$

Step 4.2

Multiply $\frac{1}{2}$ by each element of the matrix.

$x = [\begin{array}{c} \frac{1}{2} \cdot 6 & \frac{1}{2} \cdot 6 \\ \frac{1}{2} \cdot - 8 & \frac{1}{2} \cdot 10 \end{array}]$

Step 4.3

Simplify each element in the matrix.

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Step 4.3.1

Cancel the common factor of $2$ .

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Step 4.3.1.1

Factor $2$ out of $6$ .

$x = [\begin{array}{c} \frac{1}{2} \cdot (2 (3)) & \frac{1}{2} \cdot 6 \\ \frac{1}{2} \cdot - 8 & \frac{1}{2} \cdot 10 \end{array}]$

Step 4.3.1.2

Cancel the common factor.

$x = [\begin{array}{c} \frac{1}{2} \cdot (2 \cdot 3) & \frac{1}{2} \cdot 6 \\ \frac{1}{2} \cdot - 8 & \frac{1}{2} \cdot 10 \end{array}]$

Step 4.3.1.3

Rewrite the expression.

$x = [\begin{array}{c} 3 & \frac{1}{2} \cdot 6 \\ \frac{1}{2} \cdot - 8 & \frac{1}{2} \cdot 10 \end{array}]$

Step 4.3.2

Cancel the common factor of $2$ .

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Step 4.3.2.1

Factor $2$ out of $6$ .

$x = [\begin{array}{c} 3 & \frac{1}{2} \cdot (2 (3)) \\ \frac{1}{2} \cdot - 8 & \frac{1}{2} \cdot 10 \end{array}]$

Step 4.3.2.2

Cancel the common factor.

$x = [\begin{array}{c} 3 & \frac{1}{2} \cdot (2 \cdot 3) \\ \frac{1}{2} \cdot - 8 & \frac{1}{2} \cdot 10 \end{array}]$

Step 4.3.2.3

Rewrite the expression.

$x = [\begin{array}{c} 3 & 3 \\ \frac{1}{2} \cdot - 8 & \frac{1}{2} \cdot 10 \end{array}]$

Step 4.3.3

Cancel the common factor of $2$ .

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Step 4.3.3.1

Factor $2$ out of $- 8$ .

$x = [\begin{array}{c} 3 & 3 \\ \frac{1}{2} \cdot (2 (- 4)) & \frac{1}{2} \cdot 10 \end{array}]$

Step 4.3.3.2

Cancel the common factor.

$x = [\begin{array}{c} 3 & 3 \\ \frac{1}{2} \cdot (2 \cdot - 4) & \frac{1}{2} \cdot 10 \end{array}]$

Step 4.3.3.3

Rewrite the expression.

$x = [\begin{array}{c} 3 & 3 \\ - 4 & \frac{1}{2} \cdot 10 \end{array}]$

Step 4.3.4

Cancel the common factor of $2$ .

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Step 4.3.4.1

Factor $2$ out of $10$ .

$x = [\begin{array}{c} 3 & 3 \\ - 4 & \frac{1}{2} \cdot (2 (5)) \end{array}]$

Step 4.3.4.2

Cancel the common factor.

$x = [\begin{array}{c} 3 & 3 \\ - 4 & \frac{1}{2} \cdot (2 \cdot 5) \end{array}]$

Step 4.3.4.3

Rewrite the expression.

$x = [\begin{array}{c} 3 & 3 \\ - 4 & 5 \end{array}]$