Finite Math Examples

$- 11 a - b = - 3$ , $- 5 a + b + c = 9$ , $17 a - c = - 1$

Step 1

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

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Step 1.1

Add $5 a$ to both sides of the equation.

$b + c = 9 + 5 a$

$- 11 a - b = - 3$

$17 a - c = - 1$

Step 1.2

Subtract $c$ from both sides of the equation.

$b = 9 + 5 a - c$

$- 11 a - b = - 3$

$17 a - c = - 1$

$b = 9 + 5 a - c$

$- 11 a - b = - 3$

$17 a - c = - 1$

Step 2

Replace all occurrences of $b$ with $9 + 5 a - c$ in each equation.

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Step 2.1

Replace all occurrences of $b$ in $- 11 a - b = - 3$ with $9 + 5 a - c$ .

$- 11 a - (9 + 5 a - c) = - 3$

$b = 9 + 5 a - c$

$17 a - c = - 1$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $- 11 a - (9 + 5 a - c)$ .

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Step 2.2.1.1

Simplify each term.

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Step 2.2.1.1.1

Apply the distributive property.

$- 11 a - 1 \cdot 9 - (5 a) + c = - 3$

$b = 9 + 5 a - c$

$17 a - c = - 1$

Step 2.2.1.1.2

Simplify.

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Step 2.2.1.1.2.1

Multiply $- 1$ by $9$ .

$- 11 a - 9 - (5 a) + c = - 3$

$b = 9 + 5 a - c$

$17 a - c = - 1$

Step 2.2.1.1.2.2

Multiply $5$ by $- 1$ .

$- 11 a - 9 - 5 a + c = - 3$

$b = 9 + 5 a - c$

$17 a - c = - 1$

Step 2.2.1.1.2.3

Multiply $- - c$ .

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Step 2.2.1.1.2.3.1

Multiply $- 1$ by $- 1$ .

$- 11 a - 9 - 5 a + 1 c = - 3$

$b = 9 + 5 a - c$

$17 a - c = - 1$

Step 2.2.1.1.2.3.2

Multiply $c$ by $1$ .

$- 11 a - 9 - 5 a + c = - 3$

$b = 9 + 5 a - c$

$17 a - c = - 1$

$- 11 a - 9 - 5 a + c = - 3$

$b = 9 + 5 a - c$

$17 a - c = - 1$

$- 11 a - 9 - 5 a + c = - 3$

$b = 9 + 5 a - c$

$17 a - c = - 1$

$- 11 a - 9 - 5 a + c = - 3$

$b = 9 + 5 a - c$

$17 a - c = - 1$

Step 2.2.1.2

Subtract $5 a$ from $- 11 a$ .

$- 16 a - 9 + c = - 3$

$b = 9 + 5 a - c$

$17 a - c = - 1$

$- 16 a - 9 + c = - 3$

$b = 9 + 5 a - c$

$17 a - c = - 1$

$- 16 a - 9 + c = - 3$

$b = 9 + 5 a - c$

$17 a - c = - 1$

$- 16 a - 9 + c = - 3$

$b = 9 + 5 a - c$

$17 a - c = - 1$

Step 3

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

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Step 3.1

Add $16 a$ to both sides of the equation.

$- 9 + c = - 3 + 16 a$

$b = 9 + 5 a - c$

$17 a - c = - 1$

Step 3.2

Add $9$ to both sides of the equation.

$c = - 3 + 16 a + 9$

$b = 9 + 5 a - c$

$17 a - c = - 1$

Step 3.3

Add $- 3$ and $9$ .

$c = 16 a + 6$

$b = 9 + 5 a - c$

$17 a - c = - 1$

$c = 16 a + 6$

$b = 9 + 5 a - c$

$17 a - c = - 1$

Step 4

Replace all occurrences of $c$ with $16 a + 6$ in each equation.

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Step 4.1

Replace all occurrences of $c$ in $b = 9 + 5 a - c$ with $16 a + 6$ .

$b = 9 + 5 a - (16 a + 6)$

$c = 16 a + 6$

$17 a - c = - 1$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $9 + 5 a - (16 a + 6)$ .

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Step 4.2.1.1

Simplify each term.

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Step 4.2.1.1.1

Apply the distributive property.

$b = 9 + 5 a - (16 a) - 1 \cdot 6$

$c = 16 a + 6$

$17 a - c = - 1$

Step 4.2.1.1.2

Multiply $16$ by $- 1$ .

$b = 9 + 5 a - 16 a - 1 \cdot 6$

$c = 16 a + 6$

$17 a - c = - 1$

Step 4.2.1.1.3

Multiply $- 1$ by $6$ .

$b = 9 + 5 a - 16 a - 6$

$c = 16 a + 6$

$17 a - c = - 1$

$b = 9 + 5 a - 16 a - 6$

$c = 16 a + 6$

$17 a - c = - 1$

Step 4.2.1.2

Simplify by adding terms.

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Step 4.2.1.2.1

Subtract $6$ from $9$ .

$b = 5 a - 16 a + 3$

$c = 16 a + 6$

$17 a - c = - 1$

Step 4.2.1.2.2

Subtract $16 a$ from $5 a$ .

$b = - 11 a + 3$

$c = 16 a + 6$

$17 a - c = - 1$

$b = - 11 a + 3$

$c = 16 a + 6$

$17 a - c = - 1$

$b = - 11 a + 3$

$c = 16 a + 6$

$17 a - c = - 1$

$b = - 11 a + 3$

$c = 16 a + 6$

$17 a - c = - 1$

Step 4.3

Replace all occurrences of $c$ in $17 a - c = - 1$ with $16 a + 6$ .

$17 a - (16 a + 6) = - 1$

$b = - 11 a + 3$

$c = 16 a + 6$

Step 4.4

Simplify the left side.

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Step 4.4.1

Simplify $17 a - (16 a + 6)$ .

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Step 4.4.1.1

Simplify each term.

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Step 4.4.1.1.1

Apply the distributive property.

$17 a - (16 a) - 1 \cdot 6 = - 1$

$b = - 11 a + 3$

$c = 16 a + 6$

Step 4.4.1.1.2

Multiply $16$ by $- 1$ .

$17 a - 16 a - 1 \cdot 6 = - 1$

$b = - 11 a + 3$

$c = 16 a + 6$

Step 4.4.1.1.3

Multiply $- 1$ by $6$ .

$17 a - 16 a - 6 = - 1$

$b = - 11 a + 3$

$c = 16 a + 6$

$17 a - 16 a - 6 = - 1$

$b = - 11 a + 3$

$c = 16 a + 6$

Step 4.4.1.2

Subtract $16 a$ from $17 a$ .

$a - 6 = - 1$

$b = - 11 a + 3$

$c = 16 a + 6$

$a - 6 = - 1$

$b = - 11 a + 3$

$c = 16 a + 6$

$a - 6 = - 1$

$b = - 11 a + 3$

$c = 16 a + 6$

$a - 6 = - 1$

$b = - 11 a + 3$

$c = 16 a + 6$

Step 5

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

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Step 5.1

Add $6$ to both sides of the equation.

$a = - 1 + 6$

$b = - 11 a + 3$

$c = 16 a + 6$

Step 5.2

Add $- 1$ and $6$ .

$a = 5$

$b = - 11 a + 3$

$c = 16 a + 6$

$a = 5$

$b = - 11 a + 3$

$c = 16 a + 6$

Step 6

Replace all occurrences of $a$ with $5$ in each equation.

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Step 6.1

Replace all occurrences of $a$ in $b = - 11 a + 3$ with $5$ .

$b = - 11 \cdot 5 + 3$

$a = 5$

$c = 16 a + 6$

Step 6.2

Simplify the right side.

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Step 6.2.1

Simplify $- 11 \cdot 5 + 3$ .

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Step 6.2.1.1

Multiply $- 11$ by $5$ .

$b = - 55 + 3$

$a = 5$

$c = 16 a + 6$

Step 6.2.1.2

Add $- 55$ and $3$ .

$b = - 52$

$a = 5$

$c = 16 a + 6$

$b = - 52$

$a = 5$

$c = 16 a + 6$

$b = - 52$

$a = 5$

$c = 16 a + 6$

Step 6.3

Replace all occurrences of $a$ in $c = 16 a + 6$ with $5$ .

$c = 16 (5) + 6$

$b = - 52$

$a = 5$

Step 6.4

Simplify the right side.

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Step 6.4.1

Simplify $16 (5) + 6$ .

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Step 6.4.1.1

Multiply $16$ by $5$ .

$c = 80 + 6$

$b = - 52$

$a = 5$

Step 6.4.1.2

Add $80$ and $6$ .

$c = 86$

$b = - 52$

$a = 5$

$c = 86$

$b = - 52$

$a = 5$

$c = 86$

$b = - 52$

$a = 5$

$c = 86$

$b = - 52$

$a = 5$

Step 7

List all of the solutions.

$c = 86, b = - 52, a = 5$