Linear Algebra Examples

$[\begin{array}{c} 1 & 2 \\ - 3 & - 6 \end{array}] \cdot [\begin{array}{c} x \\ y \end{array}] = [\begin{array}{c} 4 \\ - 2 \end{array}]$

Step 1

Multiply $[\begin{array}{c} 1 & 2 \\ - 3 & - 6 \end{array}] [\begin{array}{c} x \\ y \end{array}]$ .

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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 $2 \times 2$ and the second matrix is $2 \times 1$ .

Step 1.2

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

$[\begin{array}{c} 1 x + 2 y \\ - 3 x - 6 y \end{array}] = [\begin{array}{c} 4 \\ - 2 \end{array}]$

Step 1.3

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

$[\begin{array}{c} x + 2 y \\ - 3 x - 6 y \end{array}] = [\begin{array}{c} 4 \\ - 2 \end{array}]$

Step 2

The matrix equation can be written as a set of equations.

$x + 2 y = 4$

$- 3 x - 6 y = - 2$

Step 3

Subtract $2 y$ from both sides of the equation.

$x = 4 - 2 y$

$- 3 x - 6 y = - 2$

Step 4

Replace all occurrences of $x$ with $4 - 2 y$ in each equation.

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Step 4.1

Replace all occurrences of $x$ in $- 3 x - 6 y = - 2$ with $4 - 2 y$ .

$- 3 (4 - 2 y) - 6 y = - 2$

$x = 4 - 2 y$

Step 4.2

Simplify the left side.

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Step 4.2.1

Simplify $- 3 (4 - 2 y) - 6 y$ .

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Step 4.2.1.1

Simplify each term.

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Step 4.2.1.1.1

Apply the distributive property.

$- 3 \cdot 4 - 3 (- 2 y) - 6 y = - 2$

$x = 4 - 2 y$

Step 4.2.1.1.2

Multiply $- 3$ by $4$ .

$- 12 - 3 (- 2 y) - 6 y = - 2$

$x = 4 - 2 y$

Step 4.2.1.1.3

Multiply $- 2$ by $- 3$ .

$- 12 + 6 y - 6 y = - 2$

$x = 4 - 2 y$

$- 12 + 6 y - 6 y = - 2$

$x = 4 - 2 y$

Step 4.2.1.2

Combine the opposite terms in $- 12 + 6 y - 6 y$ .

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Step 4.2.1.2.1

Subtract $6 y$ from $6 y$ .

$- 12 + 0 = - 2$

$x = 4 - 2 y$

Step 4.2.1.2.2

Add $- 12$ and $0$ .

$- 12 = - 2$

$x = 4 - 2 y$

$- 12 = - 2$

$x = 4 - 2 y$

$- 12 = - 2$

$x = 4 - 2 y$

$- 12 = - 2$

$x = 4 - 2 y$

$- 12 = - 2$

$x = 4 - 2 y$

Step 5

Since $- 12 = - 2$ is not true, there is no solution.

No solution