Finite Math Examples

$- 22 = 4 a - 2 b + c$ , $6 = 4 a + 2 b + c$ , $- 16 = 8 a + 2 c$

Step 1

Solve for $c$ in $- 22 = 4 a - 2 b + c$ .

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Step 1.1

Rewrite the equation as $4 a - 2 b + c = - 22$ .

$4 a - 2 b + c = - 22$

$6 = 4 a + 2 b + c$

$- 16 = 8 a + 2 c$

Step 1.2

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

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Step 1.2.1

Subtract $4 a$ from both sides of the equation.

$- 2 b + c = - 22 - 4 a$

$6 = 4 a + 2 b + c$

$- 16 = 8 a + 2 c$

Step 1.2.2

Add $2 b$ to both sides of the equation.

$c = - 22 - 4 a + 2 b$

$6 = 4 a + 2 b + c$

$- 16 = 8 a + 2 c$

$c = - 22 - 4 a + 2 b$

$6 = 4 a + 2 b + c$

$- 16 = 8 a + 2 c$

$c = - 22 - 4 a + 2 b$

$6 = 4 a + 2 b + c$

$- 16 = 8 a + 2 c$

Step 2

Replace all occurrences of $c$ with $- 22 - 4 a + 2 b$ in each equation.

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Step 2.1

Replace all occurrences of $c$ in $6 = 4 a + 2 b + c$ with $- 22 - 4 a + 2 b$ .

$6 = 4 a + 2 b - 22 - 4 a + 2 b$

$c = - 22 - 4 a + 2 b$

$- 16 = 8 a + 2 c$

Step 2.2

Simplify $6 = 4 a + 2 b - 22 - 4 a + 2 b$ .

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Step 2.2.1

Simplify the left side.

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Step 2.2.1.1

Remove parentheses.

$6 = 4 a + 2 b - 22 - 4 a + 2 b$

$c = - 22 - 4 a + 2 b$

$- 16 = 8 a + 2 c$

$6 = 4 a + 2 b - 22 - 4 a + 2 b$

$c = - 22 - 4 a + 2 b$

$- 16 = 8 a + 2 c$

Step 2.2.2

Simplify the right side.

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Step 2.2.2.1

Simplify $4 a + 2 b - 22 - 4 a + 2 b$ .

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Step 2.2.2.1.1

Combine the opposite terms in $4 a + 2 b - 22 - 4 a + 2 b$ .

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Step 2.2.2.1.1.1

Subtract $4 a$ from $4 a$ .

$6 = 2 b - 22 + 0 + 2 b$

$c = - 22 - 4 a + 2 b$

$- 16 = 8 a + 2 c$

Step 2.2.2.1.1.2

Add $2 b - 22$ and $0$ .

$6 = 2 b - 22 + 2 b$

$c = - 22 - 4 a + 2 b$

$- 16 = 8 a + 2 c$

$6 = 2 b - 22 + 2 b$

$c = - 22 - 4 a + 2 b$

$- 16 = 8 a + 2 c$

Step 2.2.2.1.2

Add $2 b$ and $2 b$ .

$6 = 4 b - 22$

$c = - 22 - 4 a + 2 b$

$- 16 = 8 a + 2 c$

$6 = 4 b - 22$

$c = - 22 - 4 a + 2 b$

$- 16 = 8 a + 2 c$

$6 = 4 b - 22$

$c = - 22 - 4 a + 2 b$

$- 16 = 8 a + 2 c$

$6 = 4 b - 22$

$c = - 22 - 4 a + 2 b$

$- 16 = 8 a + 2 c$

Step 2.3

Replace all occurrences of $c$ in $- 16 = 8 a + 2 c$ with $- 22 - 4 a + 2 b$ .

$- 16 = 8 a + 2 (- 22 - 4 a + 2 b)$

$6 = 4 b - 22$

$c = - 22 - 4 a + 2 b$

Step 2.4

Simplify the right side.

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Step 2.4.1

Simplify $8 a + 2 (- 22 - 4 a + 2 b)$ .

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Step 2.4.1.1

Simplify each term.

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Step 2.4.1.1.1

Apply the distributive property.

$- 16 = 8 a + 2 \cdot - 22 + 2 (- 4 a) + 2 (2 b)$

$6 = 4 b - 22$

$c = - 22 - 4 a + 2 b$

Step 2.4.1.1.2

Simplify.

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Step 2.4.1.1.2.1

Multiply $2$ by $- 22$ .

$- 16 = 8 a - 44 + 2 (- 4 a) + 2 (2 b)$

$6 = 4 b - 22$

$c = - 22 - 4 a + 2 b$

Step 2.4.1.1.2.2

Multiply $- 4$ by $2$ .

$- 16 = 8 a - 44 - 8 a + 2 (2 b)$

$6 = 4 b - 22$

$c = - 22 - 4 a + 2 b$

Step 2.4.1.1.2.3

Multiply $2$ by $2$ .

$- 16 = 8 a - 44 - 8 a + 4 b$

$6 = 4 b - 22$

$c = - 22 - 4 a + 2 b$

$- 16 = 8 a - 44 - 8 a + 4 b$

$6 = 4 b - 22$

$c = - 22 - 4 a + 2 b$

$- 16 = 8 a - 44 - 8 a + 4 b$

$6 = 4 b - 22$

$c = - 22 - 4 a + 2 b$

Step 2.4.1.2

Combine the opposite terms in $8 a - 44 - 8 a + 4 b$ .

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Step 2.4.1.2.1

Subtract $8 a$ from $8 a$ .

$- 16 = 0 - 44 + 4 b$

$6 = 4 b - 22$

$c = - 22 - 4 a + 2 b$

Step 2.4.1.2.2

Subtract $44$ from $0$ .

$- 16 = - 44 + 4 b$

$6 = 4 b - 22$

$c = - 22 - 4 a + 2 b$

$- 16 = - 44 + 4 b$

$6 = 4 b - 22$

$c = - 22 - 4 a + 2 b$

$- 16 = - 44 + 4 b$

$6 = 4 b - 22$

$c = - 22 - 4 a + 2 b$

$- 16 = - 44 + 4 b$

$6 = 4 b - 22$

$c = - 22 - 4 a + 2 b$

$- 16 = - 44 + 4 b$

$6 = 4 b - 22$

$c = - 22 - 4 a + 2 b$

Step 3

Solve for $b$ in $- 16 = - 44 + 4 b$ .

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Step 3.1

Rewrite the equation as $- 44 + 4 b = - 16$ .

$- 44 + 4 b = - 16$

$6 = 4 b - 22$

$c = - 22 - 4 a + 2 b$

Step 3.2

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

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Step 3.2.1

Add $44$ to both sides of the equation.

$4 b = - 16 + 44$

$6 = 4 b - 22$

$c = - 22 - 4 a + 2 b$

Step 3.2.2

Add $- 16$ and $44$ .

$4 b = 28$

$6 = 4 b - 22$

$c = - 22 - 4 a + 2 b$

$4 b = 28$

$6 = 4 b - 22$

$c = - 22 - 4 a + 2 b$

Step 3.3

Divide each term in $4 b = 28$ by $4$ and simplify.

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Step 3.3.1

Divide each term in $4 b = 28$ by $4$ .

$\frac{4 b}{4} = \frac{28}{4}$

$6 = 4 b - 22$

$c = - 22 - 4 a + 2 b$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $4$ .

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Step 3.3.2.1.1

Cancel the common factor.

$\frac{4 b}{4} = \frac{28}{4}$

$6 = 4 b - 22$

$c = - 22 - 4 a + 2 b$

Step 3.3.2.1.2

Divide $b$ by $1$ .

$b = \frac{28}{4}$

$6 = 4 b - 22$

$c = - 22 - 4 a + 2 b$

$b = \frac{28}{4}$

$6 = 4 b - 22$

$c = - 22 - 4 a + 2 b$

$b = \frac{28}{4}$

$6 = 4 b - 22$

$c = - 22 - 4 a + 2 b$

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

Divide $28$ by $4$ .

$b = 7$

$6 = 4 b - 22$

$c = - 22 - 4 a + 2 b$

$b = 7$

$6 = 4 b - 22$

$c = - 22 - 4 a + 2 b$

$b = 7$

$6 = 4 b - 22$

$c = - 22 - 4 a + 2 b$

$b = 7$

$6 = 4 b - 22$

$c = - 22 - 4 a + 2 b$

Step 4

Replace all occurrences of $b$ with $7$ in each equation.

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Step 4.1

Replace all occurrences of $b$ in $6 = 4 b - 22$ with $7$ .

$6 = 4 (7) - 22$

$b = 7$

$c = - 22 - 4 a + 2 b$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $4 (7) - 22$ .

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Step 4.2.1.1

Multiply $4$ by $7$ .

$6 = 28 - 22$

$b = 7$

$c = - 22 - 4 a + 2 b$

Step 4.2.1.2

Subtract $22$ from $28$ .

$6 = 6$

$b = 7$

$c = - 22 - 4 a + 2 b$

$6 = 6$

$b = 7$

$c = - 22 - 4 a + 2 b$

$6 = 6$

$b = 7$

$c = - 22 - 4 a + 2 b$

Step 4.3

Replace all occurrences of $b$ in $c = - 22 - 4 a + 2 b$ with $7$ .

$c = - 22 - 4 a + 2 (7)$

$6 = 6$

$b = 7$

Step 4.4

Simplify the right side.

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Step 4.4.1

Simplify $- 22 - 4 a + 2 (7)$ .

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Step 4.4.1.1

Multiply $2$ by $7$ .

$c = - 22 - 4 a + 14$

$6 = 6$

$b = 7$

Step 4.4.1.2

Add $- 22$ and $14$ .

$c = - 4 a - 8$

$6 = 6$

$b = 7$

$c = - 4 a - 8$

$6 = 6$

$b = 7$

$c = - 4 a - 8$

$6 = 6$

$b = 7$

$c = - 4 a - 8$

$6 = 6$

$b = 7$

Step 5

Remove any equations from the system that are always true.

$c = - 4 a - 8$

$b = 7$