Linear Algebra Examples

$x [\begin{array}{c} 1 \\ - 1 \end{array}] + y [\begin{array}{c} 2 \\ 1 \end{array}] = [\begin{array}{c} a \\ b \end{array}]$

Step 1

Simplify each term.

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Step 1.1

Multiply $x$ by each element of the matrix.

$[\begin{array}{c} x \cdot 1 \\ x \cdot - 1 \end{array}] + y [\begin{array}{c} 2 \\ 1 \end{array}] = [\begin{array}{c} a \\ b \end{array}]$

Step 1.2

Simplify each element in the matrix.

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Step 1.2.1

Multiply $x$ by $1$ .

$[\begin{array}{c} x \\ x \cdot - 1 \end{array}] + y [\begin{array}{c} 2 \\ 1 \end{array}] = [\begin{array}{c} a \\ b \end{array}]$

Step 1.2.2

Move $- 1$ to the left of $x$ .

$[\begin{array}{c} x \\ - 1 \cdot x \end{array}] + y [\begin{array}{c} 2 \\ 1 \end{array}] = [\begin{array}{c} a \\ b \end{array}]$

Step 1.2.3

Rewrite $- 1 x$ as $- x$ .

$[\begin{array}{c} x \\ - x \end{array}] + y [\begin{array}{c} 2 \\ 1 \end{array}] = [\begin{array}{c} a \\ b \end{array}]$

Step 1.3

Multiply $y$ by each element of the matrix.

$[\begin{array}{c} x \\ - x \end{array}] + [\begin{array}{c} y \cdot 2 \\ y \cdot 1 \end{array}] = [\begin{array}{c} a \\ b \end{array}]$

Step 1.4

Simplify each element in the matrix.

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Step 1.4.1

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

$[\begin{array}{c} x \\ - x \end{array}] + [\begin{array}{c} 2 y \\ y \cdot 1 \end{array}] = [\begin{array}{c} a \\ b \end{array}]$

Step 1.4.2

Multiply $y$ by $1$ .

$[\begin{array}{c} x \\ - x \end{array}] + [\begin{array}{c} 2 y \\ y \end{array}] = [\begin{array}{c} a \\ b \end{array}]$

Step 2

Add the corresponding elements.

$[\begin{array}{c} x + 2 y \\ - x + y \end{array}] = [\begin{array}{c} a \\ b \end{array}]$

Step 3

Write as a linear system of equations.

$x + 2 y = a$

$- x + y = b$

Step 4

Reduce the system.

$x + 2 y = a, - x + y = b$