Finite Math Examples

$[\begin{array}{c} a & b \\ c & d \end{array}] + [\begin{array}{c} - 3 & - 2 \\ 5 & - 10 \end{array}] = [\begin{array}{c} - 9 & - 1 \\ 8 & 1 \end{array}]$

Step 1

Add the corresponding elements.

$[\begin{array}{c} a - 3 & b - 2 \\ c + 5 & d - 10 \end{array}] = [\begin{array}{c} - 9 & - 1 \\ 8 & 1 \end{array}]$

Step 2

Write as a linear system of equations.

$a - 3 = - 9$

$b - 2 = - 1$

$c + 5 = 8$

$d - 10 = 1$

Step 3

Solve the system of equations.

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Step 3.1

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

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Step 3.1.1

Add $3$ to both sides of the equation.

$a = - 9 + 3$

$b - 2 = - 1$

$c + 5 = 8$

$d - 10 = 1$

Step 3.1.2

Add $- 9$ and $3$ .

$a = - 6$

$b - 2 = - 1$

$c + 5 = 8$

$d - 10 = 1$

$a = - 6$

$b - 2 = - 1$

$c + 5 = 8$

$d - 10 = 1$

Step 3.2

Replace all occurrences of $a$ with $- 6$ in each equation.

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Step 3.2.1

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

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Step 3.2.1.1

Add $2$ to both sides of the equation.

$b = - 1 + 2$

$a = - 6$

$c + 5 = 8$

$d - 10 = 1$

Step 3.2.1.2

Add $- 1$ and $2$ .

$b = 1$

$a = - 6$

$c + 5 = 8$

$d - 10 = 1$

$b = 1$

$a = - 6$

$c + 5 = 8$

$d - 10 = 1$

Step 3.2.2

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

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Step 3.2.2.1

Subtract $5$ from both sides of the equation.

$c = 8 - 5$

$b = 1$

$a = - 6$

$d - 10 = 1$

Step 3.2.2.2

Subtract $5$ from $8$ .

$c = 3$

$b = 1$

$a = - 6$

$d - 10 = 1$

$c = 3$

$b = 1$

$a = - 6$

$d - 10 = 1$

Step 3.2.3

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

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Step 3.2.3.1

Add $10$ to both sides of the equation.

$d = 1 + 10$

$c = 3$

$b = 1$

$a = - 6$

Step 3.2.3.2

Add $1$ and $10$ .

$d = 11$

$c = 3$

$b = 1$

$a = - 6$

$d = 11$

$c = 3$

$b = 1$

$a = - 6$

$d = 11$

$c = 3$

$b = 1$

$a = - 6$

Step 3.3

List all of the solutions.

$d = 11, c = 3, b = 1, a = - 6$