Finite Math Examples

$[\begin{array}{c} a & b \\ c & d \end{array}] + [\begin{array}{c} - 3 & - 10 \\ 10 & - 1 \end{array}] = [\begin{array}{c} - 2 & - 3 \\ - 6 & - 4 \end{array}]$

Step 1

Add the corresponding elements.

$[\begin{array}{c} a - 3 & b - 10 \\ c + 10 & d - 1 \end{array}] = [\begin{array}{c} - 2 & - 3 \\ - 6 & - 4 \end{array}]$

Step 2

Write as a linear system of equations.

$a - 3 = - 2$

$b - 10 = - 3$

$c + 10 = - 6$

$d - 1 = - 4$

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.

$a = - 2 + 3$

$b - 10 = - 3$

$c + 10 = - 6$

$d - 1 = - 4$

Step 3.1.2

Add $- 2$ and $3$ .

$a = 1$

$b - 10 = - 3$

$c + 10 = - 6$

$d - 1 = - 4$

$a = 1$

$b - 10 = - 3$

$c + 10 = - 6$

$d - 1 = - 4$

Step 3.2

Replace all occurrences of $a$ with $1$ 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 $10$ to both sides of the equation.

$b = - 3 + 10$

$a = 1$

$c + 10 = - 6$

$d - 1 = - 4$

Step 3.2.1.2

Add $- 3$ and $10$ .

$b = 7$

$a = 1$

$c + 10 = - 6$

$d - 1 = - 4$

$b = 7$

$a = 1$

$c + 10 = - 6$

$d - 1 = - 4$

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 $10$ from both sides of the equation.

$c = - 6 - 10$

$b = 7$

$a = 1$

$d - 1 = - 4$

Step 3.2.2.2

Subtract $10$ from $- 6$ .

$c = - 16$

$b = 7$

$a = 1$

$d - 1 = - 4$

$c = - 16$

$b = 7$

$a = 1$

$d - 1 = - 4$

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 $1$ to both sides of the equation.

$d = - 4 + 1$

$c = - 16$

$b = 7$

$a = 1$

Step 3.2.3.2

Add $- 4$ and $1$ .

$d = - 3$

$c = - 16$

$b = 7$

$a = 1$

$d = - 3$

$c = - 16$

$b = 7$

$a = 1$

$d = - 3$

$c = - 16$

$b = 7$

$a = 1$

Step 3.3

List all of the solutions.

$d = - 3, c = - 16, b = 7, a = 1$