Finite Math Examples

$[\begin{array}{c} a + 2 b & - 2 a - 3 b \\ c + 2 d & - 2 c - 3 d \end{array}] = [\begin{array}{c} 2 & - 4 \\ 5 & - 4 \end{array}]$

Step 1

Write as a linear system of equations.

$a + 2 b = 2$

$- 2 a - 3 b = - 4$

$c + 2 d = 5$

$- 2 c - 3 d = - 4$

Step 2

Solve the system of equations.

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Step 2.1

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

$a = 2 - 2 b$

$- 2 a - 3 b = - 4$

$c + 2 d = 5$

$- 2 c - 3 d = - 4$

Step 2.2

Replace all occurrences of $a$ with $2 - 2 b$ in each equation.

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Step 2.2.1

Replace all occurrences of $a$ in $- 2 a - 3 b = - 4$ with $2 - 2 b$ .

$- 2 (2 - 2 b) - 3 b = - 4$

$a = 2 - 2 b$

$c + 2 d = 5$

$- 2 c - 3 d = - 4$

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Simplify $- 2 (2 - 2 b) - 3 b$ .

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Step 2.2.2.1.1

Simplify each term.

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Step 2.2.2.1.1.1

Apply the distributive property.

$- 2 \cdot 2 - 2 (- 2 b) - 3 b = - 4$

$a = 2 - 2 b$

$c + 2 d = 5$

$- 2 c - 3 d = - 4$

Step 2.2.2.1.1.2

Multiply $- 2$ by $2$ .

$- 4 - 2 (- 2 b) - 3 b = - 4$

$a = 2 - 2 b$

$c + 2 d = 5$

$- 2 c - 3 d = - 4$

Step 2.2.2.1.1.3

Multiply $- 2$ by $- 2$ .

$- 4 + 4 b - 3 b = - 4$

$a = 2 - 2 b$

$c + 2 d = 5$

$- 2 c - 3 d = - 4$

$- 4 + 4 b - 3 b = - 4$

$a = 2 - 2 b$

$c + 2 d = 5$

$- 2 c - 3 d = - 4$

Step 2.2.2.1.2

Subtract $3 b$ from $4 b$ .

$- 4 + b = - 4$

$a = 2 - 2 b$

$c + 2 d = 5$

$- 2 c - 3 d = - 4$

$- 4 + b = - 4$

$a = 2 - 2 b$

$c + 2 d = 5$

$- 2 c - 3 d = - 4$

$- 4 + b = - 4$

$a = 2 - 2 b$

$c + 2 d = 5$

$- 2 c - 3 d = - 4$

$- 4 + b = - 4$

$a = 2 - 2 b$

$c + 2 d = 5$

$- 2 c - 3 d = - 4$

Step 2.3

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

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Step 2.3.1

Add $4$ to both sides of the equation.

$b = - 4 + 4$

$a = 2 - 2 b$

$c + 2 d = 5$

$- 2 c - 3 d = - 4$

Step 2.3.2

Add $- 4$ and $4$ .

$b = 0$

$a = 2 - 2 b$

$c + 2 d = 5$

$- 2 c - 3 d = - 4$

$b = 0$

$a = 2 - 2 b$

$c + 2 d = 5$

$- 2 c - 3 d = - 4$

Step 2.4

Replace all occurrences of $b$ with $0$ in each equation.

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Step 2.4.1

Replace all occurrences of $b$ in $a = 2 - 2 b$ with $0$ .

$a = 2 - 2 \cdot 0$

$b = 0$

$c + 2 d = 5$

$- 2 c - 3 d = - 4$

Step 2.4.2

Simplify the right side.

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Step 2.4.2.1

Simplify $2 - 2 \cdot 0$ .

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Step 2.4.2.1.1

Multiply $- 2$ by $0$ .

$a = 2 + 0$

$b = 0$

$c + 2 d = 5$

$- 2 c - 3 d = - 4$

Step 2.4.2.1.2

Add $2$ and $0$ .

$a = 2$

$b = 0$

$c + 2 d = 5$

$- 2 c - 3 d = - 4$

$a = 2$

$b = 0$

$c + 2 d = 5$

$- 2 c - 3 d = - 4$

$a = 2$

$b = 0$

$c + 2 d = 5$

$- 2 c - 3 d = - 4$

$a = 2$

$b = 0$

$c + 2 d = 5$

$- 2 c - 3 d = - 4$

Step 2.5

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

$c = 5 - 2 d$

$a = 2$

$b = 0$

$- 2 c - 3 d = - 4$

Step 2.6

Replace all occurrences of $c$ with $5 - 2 d$ in each equation.

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Step 2.6.1

Replace all occurrences of $c$ in $- 2 c - 3 d = - 4$ with $5 - 2 d$ .

$- 2 (5 - 2 d) - 3 d = - 4$

$c = 5 - 2 d$

$a = 2$

$b = 0$

Step 2.6.2

Simplify the left side.

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Step 2.6.2.1

Simplify $- 2 (5 - 2 d) - 3 d$ .

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Step 2.6.2.1.1

Simplify each term.

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Step 2.6.2.1.1.1

Apply the distributive property.

$- 2 \cdot 5 - 2 (- 2 d) - 3 d = - 4$

$c = 5 - 2 d$

$a = 2$

$b = 0$

Step 2.6.2.1.1.2

Multiply $- 2$ by $5$ .

$- 10 - 2 (- 2 d) - 3 d = - 4$

$c = 5 - 2 d$

$a = 2$

$b = 0$

Step 2.6.2.1.1.3

Multiply $- 2$ by $- 2$ .

$- 10 + 4 d - 3 d = - 4$

$c = 5 - 2 d$

$a = 2$

$b = 0$

$- 10 + 4 d - 3 d = - 4$

$c = 5 - 2 d$

$a = 2$

$b = 0$

Step 2.6.2.1.2

Subtract $3 d$ from $4 d$ .

$- 10 + d = - 4$

$c = 5 - 2 d$

$a = 2$

$b = 0$

$- 10 + d = - 4$

$c = 5 - 2 d$

$a = 2$

$b = 0$

$- 10 + d = - 4$

$c = 5 - 2 d$

$a = 2$

$b = 0$

$- 10 + d = - 4$

$c = 5 - 2 d$

$a = 2$

$b = 0$

Step 2.7

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

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Step 2.7.1

Add $10$ to both sides of the equation.

$d = - 4 + 10$

$c = 5 - 2 d$

$a = 2$

$b = 0$

Step 2.7.2

Add $- 4$ and $10$ .

$d = 6$

$c = 5 - 2 d$

$a = 2$

$b = 0$

$d = 6$

$c = 5 - 2 d$

$a = 2$

$b = 0$

Step 2.8

Replace all occurrences of $d$ with $6$ in each equation.

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Step 2.8.1

Replace all occurrences of $d$ in $c = 5 - 2 d$ with $6$ .

$c = 5 - 2 \cdot 6$

$d = 6$

$a = 2$

$b = 0$

Step 2.8.2

Simplify the right side.

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Step 2.8.2.1

Simplify $5 - 2 \cdot 6$ .

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Step 2.8.2.1.1

Multiply $- 2$ by $6$ .

$c = 5 - 12$

$d = 6$

$a = 2$

$b = 0$

Step 2.8.2.1.2

Subtract $12$ from $5$ .

$c = - 7$

$d = 6$

$a = 2$

$b = 0$

$c = - 7$

$d = 6$

$a = 2$

$b = 0$

$c = - 7$

$d = 6$

$a = 2$

$b = 0$

$c = - 7$

$d = 6$

$a = 2$

$b = 0$

Step 2.9

List all of the solutions.

$c = - 7, d = 6, a = 2, b = 0$