Finite Math Examples

$4 a + 3 b + c = - 34$ , $a - 3 b + 2 c = - 4$ , $11 a - 2 b + 3 c = - 54$

Step 1

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

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Step 1.1

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

$3 b + c = - 34 - 4 a$

$a - 3 b + 2 c = - 4$

$11 a - 2 b + 3 c = - 54$

Step 1.2

Subtract $3 b$ from both sides of the equation.

$c = - 34 - 4 a - 3 b$

$a - 3 b + 2 c = - 4$

$11 a - 2 b + 3 c = - 54$

$c = - 34 - 4 a - 3 b$

$a - 3 b + 2 c = - 4$

$11 a - 2 b + 3 c = - 54$

Step 2

Replace all occurrences of $c$ with $- 34 - 4 a - 3 b$ in each equation.

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Step 2.1

Replace all occurrences of $c$ in $a - 3 b + 2 c = - 4$ with $- 34 - 4 a - 3 b$ .

$a - 3 b + 2 (- 34 - 4 a - 3 b) = - 4$

$c = - 34 - 4 a - 3 b$

$11 a - 2 b + 3 c = - 54$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $a - 3 b + 2 (- 34 - 4 a - 3 b)$ .

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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.

$a - 3 b + 2 \cdot - 34 + 2 (- 4 a) + 2 (- 3 b) = - 4$

$c = - 34 - 4 a - 3 b$

$11 a - 2 b + 3 c = - 54$

Step 2.2.1.1.2

Simplify.

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Step 2.2.1.1.2.1

Multiply $2$ by $- 34$ .

$a - 3 b - 68 + 2 (- 4 a) + 2 (- 3 b) = - 4$

$c = - 34 - 4 a - 3 b$

$11 a - 2 b + 3 c = - 54$

Step 2.2.1.1.2.2

Multiply $- 4$ by $2$ .

$a - 3 b - 68 - 8 a + 2 (- 3 b) = - 4$

$c = - 34 - 4 a - 3 b$

$11 a - 2 b + 3 c = - 54$

Step 2.2.1.1.2.3

Multiply $- 3$ by $2$ .

$a - 3 b - 68 - 8 a - 6 b = - 4$

$c = - 34 - 4 a - 3 b$

$11 a - 2 b + 3 c = - 54$

$a - 3 b - 68 - 8 a - 6 b = - 4$

$c = - 34 - 4 a - 3 b$

$11 a - 2 b + 3 c = - 54$

$a - 3 b - 68 - 8 a - 6 b = - 4$

$c = - 34 - 4 a - 3 b$

$11 a - 2 b + 3 c = - 54$

Step 2.2.1.2

Simplify by adding terms.

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Step 2.2.1.2.1

Subtract $8 a$ from $a$ .

$- 7 a - 3 b - 68 - 6 b = - 4$

$c = - 34 - 4 a - 3 b$

$11 a - 2 b + 3 c = - 54$

Step 2.2.1.2.2

Subtract $6 b$ from $- 3 b$ .

$- 7 a - 9 b - 68 = - 4$

$c = - 34 - 4 a - 3 b$

$11 a - 2 b + 3 c = - 54$

$- 7 a - 9 b - 68 = - 4$

$c = - 34 - 4 a - 3 b$

$11 a - 2 b + 3 c = - 54$

$- 7 a - 9 b - 68 = - 4$

$c = - 34 - 4 a - 3 b$

$11 a - 2 b + 3 c = - 54$

$- 7 a - 9 b - 68 = - 4$

$c = - 34 - 4 a - 3 b$

$11 a - 2 b + 3 c = - 54$

Step 2.3

Replace all occurrences of $c$ in $11 a - 2 b + 3 c = - 54$ with $- 34 - 4 a - 3 b$ .

$11 a - 2 b + 3 (- 34 - 4 a - 3 b) = - 54$

$- 7 a - 9 b - 68 = - 4$

$c = - 34 - 4 a - 3 b$

Step 2.4

Simplify the left side.

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Step 2.4.1

Simplify $11 a - 2 b + 3 (- 34 - 4 a - 3 b)$ .

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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.

$11 a - 2 b + 3 \cdot - 34 + 3 (- 4 a) + 3 (- 3 b) = - 54$

$- 7 a - 9 b - 68 = - 4$

$c = - 34 - 4 a - 3 b$

Step 2.4.1.1.2

Simplify.

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Step 2.4.1.1.2.1

Multiply $3$ by $- 34$ .

$11 a - 2 b - 102 + 3 (- 4 a) + 3 (- 3 b) = - 54$

$- 7 a - 9 b - 68 = - 4$

$c = - 34 - 4 a - 3 b$

Step 2.4.1.1.2.2

Multiply $- 4$ by $3$ .

$11 a - 2 b - 102 - 12 a + 3 (- 3 b) = - 54$

$- 7 a - 9 b - 68 = - 4$

$c = - 34 - 4 a - 3 b$

Step 2.4.1.1.2.3

Multiply $- 3$ by $3$ .

$11 a - 2 b - 102 - 12 a - 9 b = - 54$

$- 7 a - 9 b - 68 = - 4$

$c = - 34 - 4 a - 3 b$

$11 a - 2 b - 102 - 12 a - 9 b = - 54$

$- 7 a - 9 b - 68 = - 4$

$c = - 34 - 4 a - 3 b$

$11 a - 2 b - 102 - 12 a - 9 b = - 54$

$- 7 a - 9 b - 68 = - 4$

$c = - 34 - 4 a - 3 b$

Step 2.4.1.2

Simplify by adding terms.

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Step 2.4.1.2.1

Subtract $12 a$ from $11 a$ .

$- a - 2 b - 102 - 9 b = - 54$

$- 7 a - 9 b - 68 = - 4$

$c = - 34 - 4 a - 3 b$

Step 2.4.1.2.2

Subtract $9 b$ from $- 2 b$ .

$- a - 11 b - 102 = - 54$

$- 7 a - 9 b - 68 = - 4$

$c = - 34 - 4 a - 3 b$

$- a - 11 b - 102 = - 54$

$- 7 a - 9 b - 68 = - 4$

$c = - 34 - 4 a - 3 b$

$- a - 11 b - 102 = - 54$

$- 7 a - 9 b - 68 = - 4$

$c = - 34 - 4 a - 3 b$

$- a - 11 b - 102 = - 54$

$- 7 a - 9 b - 68 = - 4$

$c = - 34 - 4 a - 3 b$

$- a - 11 b - 102 = - 54$

$- 7 a - 9 b - 68 = - 4$

$c = - 34 - 4 a - 3 b$

Step 3

Solve for $a$ in $- a - 11 b - 102 = - 54$ .

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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 $11 b$ to both sides of the equation.

$- a - 102 = - 54 + 11 b$

$- 7 a - 9 b - 68 = - 4$

$c = - 34 - 4 a - 3 b$

Step 3.1.2

Add $102$ to both sides of the equation.

$- a = - 54 + 11 b + 102$

$- 7 a - 9 b - 68 = - 4$

$c = - 34 - 4 a - 3 b$

Step 3.1.3

Add $- 54$ and $102$ .

$- a = 11 b + 48$

$- 7 a - 9 b - 68 = - 4$

$c = - 34 - 4 a - 3 b$

$- a = 11 b + 48$

$- 7 a - 9 b - 68 = - 4$

$c = - 34 - 4 a - 3 b$

Step 3.2

Divide each term in $- a = 11 b + 48$ by $- 1$ and simplify.

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Step 3.2.1

Divide each term in $- a = 11 b + 48$ by $- 1$ .

$\frac{- a}{- 1} = \frac{11 b}{- 1} + \frac{48}{- 1}$

$- 7 a - 9 b - 68 = - 4$

$c = - 34 - 4 a - 3 b$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Dividing two negative values results in a positive value.

$\frac{a}{1} = \frac{11 b}{- 1} + \frac{48}{- 1}$

$- 7 a - 9 b - 68 = - 4$

$c = - 34 - 4 a - 3 b$

Step 3.2.2.2

Divide $a$ by $1$ .

$a = \frac{11 b}{- 1} + \frac{48}{- 1}$

$- 7 a - 9 b - 68 = - 4$

$c = - 34 - 4 a - 3 b$

$a = \frac{11 b}{- 1} + \frac{48}{- 1}$

$- 7 a - 9 b - 68 = - 4$

$c = - 34 - 4 a - 3 b$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Simplify each term.

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Step 3.2.3.1.1

Move the negative one from the denominator of $\frac{11 b}{- 1}$ .

$a = - 1 \cdot (11 b) + \frac{48}{- 1}$

$- 7 a - 9 b - 68 = - 4$

$c = - 34 - 4 a - 3 b$

Step 3.2.3.1.2

Rewrite $- 1 \cdot (11 b)$ as $- (11 b)$ .

$a = - (11 b) + \frac{48}{- 1}$

$- 7 a - 9 b - 68 = - 4$

$c = - 34 - 4 a - 3 b$

Step 3.2.3.1.3

Multiply $11$ by $- 1$ .

$a = - 11 b + \frac{48}{- 1}$

$- 7 a - 9 b - 68 = - 4$

$c = - 34 - 4 a - 3 b$

Step 3.2.3.1.4

Divide $48$ by $- 1$ .

$a = - 11 b - 48$

$- 7 a - 9 b - 68 = - 4$

$c = - 34 - 4 a - 3 b$

$a = - 11 b - 48$

$- 7 a - 9 b - 68 = - 4$

$c = - 34 - 4 a - 3 b$

$a = - 11 b - 48$

$- 7 a - 9 b - 68 = - 4$

$c = - 34 - 4 a - 3 b$

$a = - 11 b - 48$

$- 7 a - 9 b - 68 = - 4$

$c = - 34 - 4 a - 3 b$

$a = - 11 b - 48$

$- 7 a - 9 b - 68 = - 4$

$c = - 34 - 4 a - 3 b$

Step 4

Replace all occurrences of $a$ with $- 11 b - 48$ in each equation.

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Step 4.1

Replace all occurrences of $a$ in $- 7 a - 9 b - 68 = - 4$ with $- 11 b - 48$ .

$- 7 (- 11 b - 48) - 9 b - 68 = - 4$

$a = - 11 b - 48$

$c = - 34 - 4 a - 3 b$

Step 4.2

Simplify the left side.

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Step 4.2.1

Simplify $- 7 (- 11 b - 48) - 9 b - 68$ .

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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.

$- 7 (- 11 b) - 7 \cdot - 48 - 9 b - 68 = - 4$

$a = - 11 b - 48$

$c = - 34 - 4 a - 3 b$

Step 4.2.1.1.2

Multiply $- 11$ by $- 7$ .

$77 b - 7 \cdot - 48 - 9 b - 68 = - 4$

$a = - 11 b - 48$

$c = - 34 - 4 a - 3 b$

Step 4.2.1.1.3

Multiply $- 7$ by $- 48$ .

$77 b + 336 - 9 b - 68 = - 4$

$a = - 11 b - 48$

$c = - 34 - 4 a - 3 b$

$77 b + 336 - 9 b - 68 = - 4$

$a = - 11 b - 48$

$c = - 34 - 4 a - 3 b$

Step 4.2.1.2

Simplify by adding terms.

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Step 4.2.1.2.1

Subtract $9 b$ from $77 b$ .

$68 b + 336 - 68 = - 4$

$a = - 11 b - 48$

$c = - 34 - 4 a - 3 b$

Step 4.2.1.2.2

Subtract $68$ from $336$ .

$68 b + 268 = - 4$

$a = - 11 b - 48$

$c = - 34 - 4 a - 3 b$

$68 b + 268 = - 4$

$a = - 11 b - 48$

$c = - 34 - 4 a - 3 b$

$68 b + 268 = - 4$

$a = - 11 b - 48$

$c = - 34 - 4 a - 3 b$

$68 b + 268 = - 4$

$a = - 11 b - 48$

$c = - 34 - 4 a - 3 b$

Step 4.3

Replace all occurrences of $a$ in $c = - 34 - 4 a - 3 b$ with $- 11 b - 48$ .

$c = - 34 - 4 (- 11 b - 48) - 3 b$

$68 b + 268 = - 4$

$a = - 11 b - 48$

Step 4.4

Simplify the right side.

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Step 4.4.1

Simplify $- 34 - 4 (- 11 b - 48) - 3 b$ .

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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.

$c = - 34 - 4 (- 11 b) - 4 \cdot - 48 - 3 b$

$68 b + 268 = - 4$

$a = - 11 b - 48$

Step 4.4.1.1.2

Multiply $- 11$ by $- 4$ .

$c = - 34 + 44 b - 4 \cdot - 48 - 3 b$

$68 b + 268 = - 4$

$a = - 11 b - 48$

Step 4.4.1.1.3

Multiply $- 4$ by $- 48$ .

$c = - 34 + 44 b + 192 - 3 b$

$68 b + 268 = - 4$

$a = - 11 b - 48$

$c = - 34 + 44 b + 192 - 3 b$

$68 b + 268 = - 4$

$a = - 11 b - 48$

Step 4.4.1.2

Simplify by adding terms.

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Step 4.4.1.2.1

Add $- 34$ and $192$ .

$c = 44 b + 158 - 3 b$

$68 b + 268 = - 4$

$a = - 11 b - 48$

Step 4.4.1.2.2

Subtract $3 b$ from $44 b$ .

$c = 41 b + 158$

$68 b + 268 = - 4$

$a = - 11 b - 48$

$c = 41 b + 158$

$68 b + 268 = - 4$

$a = - 11 b - 48$

$c = 41 b + 158$

$68 b + 268 = - 4$

$a = - 11 b - 48$

$c = 41 b + 158$

$68 b + 268 = - 4$

$a = - 11 b - 48$

$c = 41 b + 158$

$68 b + 268 = - 4$

$a = - 11 b - 48$

Step 5

Solve for $b$ in $68 b + 268 = - 4$ .

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Step 5.1

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

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Step 5.1.1

Subtract $268$ from both sides of the equation.

$68 b = - 4 - 268$

$c = 41 b + 158$

$a = - 11 b - 48$

Step 5.1.2

Subtract $268$ from $- 4$ .

$68 b = - 272$

$c = 41 b + 158$

$a = - 11 b - 48$

$68 b = - 272$

$c = 41 b + 158$

$a = - 11 b - 48$

Step 5.2

Divide each term in $68 b = - 272$ by $68$ and simplify.

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Step 5.2.1

Divide each term in $68 b = - 272$ by $68$ .

$\frac{68 b}{68} = \frac{- 272}{68}$

$c = 41 b + 158$

$a = - 11 b - 48$

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Cancel the common factor of $68$ .

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Step 5.2.2.1.1

Cancel the common factor.

$\frac{68 b}{68} = \frac{- 272}{68}$

$c = 41 b + 158$

$a = - 11 b - 48$

Step 5.2.2.1.2

Divide $b$ by $1$ .

$b = \frac{- 272}{68}$

$c = 41 b + 158$

$a = - 11 b - 48$

$b = \frac{- 272}{68}$

$c = 41 b + 158$

$a = - 11 b - 48$

$b = \frac{- 272}{68}$

$c = 41 b + 158$

$a = - 11 b - 48$

Step 5.2.3

Simplify the right side.

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Step 5.2.3.1

Divide $- 272$ by $68$ .

$b = - 4$

$c = 41 b + 158$

$a = - 11 b - 48$

$b = - 4$

$c = 41 b + 158$

$a = - 11 b - 48$

$b = - 4$

$c = 41 b + 158$

$a = - 11 b - 48$

$b = - 4$

$c = 41 b + 158$

$a = - 11 b - 48$

Step 6

Replace all occurrences of $b$ with $- 4$ in each equation.

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Step 6.1

Replace all occurrences of $b$ in $c = 41 b + 158$ with $- 4$ .

$c = 41 (- 4) + 158$

$b = - 4$

$a = - 11 b - 48$

Step 6.2

Simplify the right side.

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Step 6.2.1

Simplify $41 (- 4) + 158$ .

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Step 6.2.1.1

Multiply $41$ by $- 4$ .

$c = - 164 + 158$

$b = - 4$

$a = - 11 b - 48$

Step 6.2.1.2

Add $- 164$ and $158$ .

$c = - 6$

$b = - 4$

$a = - 11 b - 48$

$c = - 6$

$b = - 4$

$a = - 11 b - 48$

$c = - 6$

$b = - 4$

$a = - 11 b - 48$

Step 6.3

Replace all occurrences of $b$ in $a = - 11 b - 48$ with $- 4$ .

$a = - 11 \cdot - 4 - 48$

$c = - 6$

$b = - 4$

Step 6.4

Simplify the right side.

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Step 6.4.1

Simplify $- 11 \cdot - 4 - 48$ .

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Step 6.4.1.1

Multiply $- 11$ by $- 4$ .

$a = 44 - 48$

$c = - 6$

$b = - 4$

Step 6.4.1.2

Subtract $48$ from $44$ .

$a = - 4$

$c = - 6$

$b = - 4$

$a = - 4$

$c = - 6$

$b = - 4$

$a = - 4$

$c = - 6$

$b = - 4$

$a = - 4$

$c = - 6$

$b = - 4$

Step 7

List all of the solutions.

$a = - 4, c = - 6, b = - 4$