Algebra Examples

$[\begin{array}{c} - 20 & 16 \\ 4 & 0 \\ 9 & - 6 \end{array}] - [\begin{array}{c} - 5 & 1 \\ - 6 & 7 \\ 0 & 11 \end{array}] = [\begin{array}{c} x & y \\ 10 & - 7 \\ 9 & - 17 \end{array}]$

Step 1

Subtract the corresponding elements.

$[\begin{array}{c} - 20 + 5 & 16 - 1 \\ 4 + 6 & 0 - 7 \\ 9 - 0 & - 6 - 11 \end{array}] = [\begin{array}{c} x & y \\ 10 & - 7 \\ 9 & - 17 \end{array}]$

Step 2

Simplify each element.

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Step 2.1

Add $- 20$ and $5$ .

$[\begin{array}{c} - 15 & 16 - 1 \\ 4 + 6 & 0 - 7 \\ 9 - 0 & - 6 - 11 \end{array}] = [\begin{array}{c} x & y \\ 10 & - 7 \\ 9 & - 17 \end{array}]$

Step 2.2

Subtract $1$ from $16$ .

$[\begin{array}{c} - 15 & 15 \\ 4 + 6 & 0 - 7 \\ 9 - 0 & - 6 - 11 \end{array}] = [\begin{array}{c} x & y \\ 10 & - 7 \\ 9 & - 17 \end{array}]$

Step 2.3

Add $4$ and $6$ .

$[\begin{array}{c} - 15 & 15 \\ 10 & 0 - 7 \\ 9 - 0 & - 6 - 11 \end{array}] = [\begin{array}{c} x & y \\ 10 & - 7 \\ 9 & - 17 \end{array}]$

Step 2.4

Subtract $7$ from $0$ .

$[\begin{array}{c} - 15 & 15 \\ 10 & - 7 \\ 9 - 0 & - 6 - 11 \end{array}] = [\begin{array}{c} x & y \\ 10 & - 7 \\ 9 & - 17 \end{array}]$

Step 2.5

Subtract $0$ from $9$ .

$[\begin{array}{c} - 15 & 15 \\ 10 & - 7 \\ 9 & - 6 - 11 \end{array}] = [\begin{array}{c} x & y \\ 10 & - 7 \\ 9 & - 17 \end{array}]$

Step 2.6

Subtract $11$ from $- 6$ .

$[\begin{array}{c} - 15 & 15 \\ 10 & - 7 \\ 9 & - 17 \end{array}] = [\begin{array}{c} x & y \\ 10 & - 7 \\ 9 & - 17 \end{array}]$

Step 3

Write as a linear system of equations.

$- 15 = x$

$15 = y$

$10 = 10$

$- 7 = - 7$

$9 = 9$

$- 17 = - 17$

Step 4

Since $- 17 = - 17$ is always true, the matrix equation has infinite solutions.

Infinite solutions satisfying the system.