Finite Math Examples

$a + b - c = 2$ , $2 a - 3 c = 11$ , $2 b + 4 c = 9$

Step 1

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

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Step 1.1

Subtract $b$ from both sides of the equation.

$a - c = 2 - b$

$2 a - 3 c = 11$

$2 b + 4 c = 9$

Step 1.2

Add $c$ to both sides of the equation.

$a = 2 - b + c$

$2 a - 3 c = 11$

$2 b + 4 c = 9$

$a = 2 - b + c$

$2 a - 3 c = 11$

$2 b + 4 c = 9$

Step 2

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

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Step 2.1

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

$2 (2 - b + c) - 3 c = 11$

$a = 2 - b + c$

$2 b + 4 c = 9$

Step 2.2

Simplify the left side.

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Step 2.2.1

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

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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.

$2 \cdot 2 + 2 (- b) + 2 c - 3 c = 11$

$a = 2 - b + c$

$2 b + 4 c = 9$

Step 2.2.1.1.2

Simplify.

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Step 2.2.1.1.2.1

Multiply $2$ by $2$ .

$4 + 2 (- b) + 2 c - 3 c = 11$

$a = 2 - b + c$

$2 b + 4 c = 9$

Step 2.2.1.1.2.2

Multiply $- 1$ by $2$ .

$4 - 2 b + 2 c - 3 c = 11$

$a = 2 - b + c$

$2 b + 4 c = 9$

$4 - 2 b + 2 c - 3 c = 11$

$a = 2 - b + c$

$2 b + 4 c = 9$

$4 - 2 b + 2 c - 3 c = 11$

$a = 2 - b + c$

$2 b + 4 c = 9$

Step 2.2.1.2

Subtract $3 c$ from $2 c$ .

$4 - 2 b - c = 11$

$a = 2 - b + c$

$2 b + 4 c = 9$

$4 - 2 b - c = 11$

$a = 2 - b + c$

$2 b + 4 c = 9$

$4 - 2 b - c = 11$

$a = 2 - b + c$

$2 b + 4 c = 9$

$4 - 2 b - c = 11$

$a = 2 - b + c$

$2 b + 4 c = 9$

Step 3

Solve for $b$ in $4 - 2 b - c = 11$ .

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Step 3.1

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

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Step 3.1.1

Subtract $4$ from both sides of the equation.

$- 2 b - c = 11 - 4$

$a = 2 - b + c$

$2 b + 4 c = 9$

Step 3.1.2

Add $c$ to both sides of the equation.

$- 2 b = 11 - 4 + c$

$a = 2 - b + c$

$2 b + 4 c = 9$

Step 3.1.3

Subtract $4$ from $11$ .

$- 2 b = 7 + c$

$a = 2 - b + c$

$2 b + 4 c = 9$

$- 2 b = 7 + c$

$a = 2 - b + c$

$2 b + 4 c = 9$

Step 3.2

Divide each term in $- 2 b = 7 + c$ by $- 2$ and simplify.

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Step 3.2.1

Divide each term in $- 2 b = 7 + c$ by $- 2$ .

$\frac{- 2 b}{- 2} = \frac{7}{- 2} + \frac{c}{- 2}$

$a = 2 - b + c$

$2 b + 4 c = 9$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $- 2$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{- 2 b}{- 2} = \frac{7}{- 2} + \frac{c}{- 2}$

$a = 2 - b + c$

$2 b + 4 c = 9$

Step 3.2.2.1.2

Divide $b$ by $1$ .

$b = \frac{7}{- 2} + \frac{c}{- 2}$

$a = 2 - b + c$

$2 b + 4 c = 9$

$b = \frac{7}{- 2} + \frac{c}{- 2}$

$a = 2 - b + c$

$2 b + 4 c = 9$

$b = \frac{7}{- 2} + \frac{c}{- 2}$

$a = 2 - b + c$

$2 b + 4 c = 9$

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 in front of the fraction.

$b = - \frac{7}{2} + \frac{c}{- 2}$

$a = 2 - b + c$

$2 b + 4 c = 9$

Step 3.2.3.1.2

Move the negative in front of the fraction.

$b = - \frac{7}{2} - \frac{c}{2}$

$a = 2 - b + c$

$2 b + 4 c = 9$

$b = - \frac{7}{2} - \frac{c}{2}$

$a = 2 - b + c$

$2 b + 4 c = 9$

$b = - \frac{7}{2} - \frac{c}{2}$

$a = 2 - b + c$

$2 b + 4 c = 9$

$b = - \frac{7}{2} - \frac{c}{2}$

$a = 2 - b + c$

$2 b + 4 c = 9$

$b = - \frac{7}{2} - \frac{c}{2}$

$a = 2 - b + c$

$2 b + 4 c = 9$

Step 4

Replace all occurrences of $b$ with $- \frac{7}{2} - \frac{c}{2}$ in each equation.

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Step 4.1

Replace all occurrences of $b$ in $a = 2 - b + c$ with $- \frac{7}{2} - \frac{c}{2}$ .

$a = 2 - (- \frac{7}{2} - \frac{c}{2}) + c$

$b = - \frac{7}{2} - \frac{c}{2}$

$2 b + 4 c = 9$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $2 - (- \frac{7}{2} - \frac{c}{2}) + c$ .

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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.

$a = 2 + \frac{7}{2} + \frac{c}{2} + c$

$b = - \frac{7}{2} - \frac{c}{2}$

$2 b + 4 c = 9$

Step 4.2.1.1.2

Multiply $- - \frac{7}{2}$ .

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Step 4.2.1.1.2.1

Multiply $- 1$ by $- 1$ .

$a = 2 + 1 (\frac{7}{2}) + \frac{c}{2} + c$

$b = - \frac{7}{2} - \frac{c}{2}$

$2 b + 4 c = 9$

Step 4.2.1.1.2.2

Multiply $\frac{7}{2}$ by $1$ .

$a = 2 + \frac{7}{2} + \frac{c}{2} + c$

$b = - \frac{7}{2} - \frac{c}{2}$

$2 b + 4 c = 9$

$a = 2 + \frac{7}{2} + \frac{c}{2} + c$

$b = - \frac{7}{2} - \frac{c}{2}$

$2 b + 4 c = 9$

Step 4.2.1.1.3

Multiply $- - \frac{c}{2}$ .

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Step 4.2.1.1.3.1

Multiply $- 1$ by $- 1$ .

$a = 2 + \frac{7}{2} + 1 (\frac{c}{2}) + c$

$b = - \frac{7}{2} - \frac{c}{2}$

$2 b + 4 c = 9$

Step 4.2.1.1.3.2

Multiply $\frac{c}{2}$ by $1$ .

$a = 2 + \frac{7}{2} + \frac{c}{2} + c$

$b = - \frac{7}{2} - \frac{c}{2}$

$2 b + 4 c = 9$

$a = 2 + \frac{7}{2} + \frac{c}{2} + c$

$b = - \frac{7}{2} - \frac{c}{2}$

$2 b + 4 c = 9$

$a = 2 + \frac{7}{2} + \frac{c}{2} + c$

$b = - \frac{7}{2} - \frac{c}{2}$

$2 b + 4 c = 9$

Step 4.2.1.2

To write $2$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$a = 2 \cdot \frac{2}{2} + \frac{7}{2} + \frac{c}{2} + c$

$b = - \frac{7}{2} - \frac{c}{2}$

$2 b + 4 c = 9$

Step 4.2.1.3

Combine $2$ and $\frac{2}{2}$ .

$a = \frac{2 \cdot 2}{2} + \frac{7}{2} + \frac{c}{2} + c$

$b = - \frac{7}{2} - \frac{c}{2}$

$2 b + 4 c = 9$

Step 4.2.1.4

Combine the numerators over the common denominator.

$a = \frac{2 \cdot 2 + 7}{2} + \frac{c}{2} + c$

$b = - \frac{7}{2} - \frac{c}{2}$

$2 b + 4 c = 9$

Step 4.2.1.5

Simplify the numerator.

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Step 4.2.1.5.1

Multiply $2$ by $2$ .

$a = \frac{4 + 7}{2} + \frac{c}{2} + c$

$b = - \frac{7}{2} - \frac{c}{2}$

$2 b + 4 c = 9$

Step 4.2.1.5.2

Add $4$ and $7$ .

$a = \frac{11}{2} + \frac{c}{2} + c$

$b = - \frac{7}{2} - \frac{c}{2}$

$2 b + 4 c = 9$

$a = \frac{11}{2} + \frac{c}{2} + c$

$b = - \frac{7}{2} - \frac{c}{2}$

$2 b + 4 c = 9$

Step 4.2.1.6

To write $c$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$a = \frac{11}{2} + \frac{c}{2} + c \cdot \frac{2}{2}$

$b = - \frac{7}{2} - \frac{c}{2}$

$2 b + 4 c = 9$

Step 4.2.1.7

Simplify terms.

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Step 4.2.1.7.1

Combine $c$ and $\frac{2}{2}$ .

$a = \frac{11}{2} + \frac{c}{2} + \frac{c \cdot 2}{2}$

$b = - \frac{7}{2} - \frac{c}{2}$

$2 b + 4 c = 9$

Step 4.2.1.7.2

Combine the numerators over the common denominator.

$a = \frac{11}{2} + \frac{c + c \cdot 2}{2}$

$b = - \frac{7}{2} - \frac{c}{2}$

$2 b + 4 c = 9$

Step 4.2.1.7.3

Combine the numerators over the common denominator.

$a = \frac{11 + c + c \cdot 2}{2}$

$b = - \frac{7}{2} - \frac{c}{2}$

$2 b + 4 c = 9$

$a = \frac{11 + c + c \cdot 2}{2}$

$b = - \frac{7}{2} - \frac{c}{2}$

$2 b + 4 c = 9$

Step 4.2.1.8

Move $2$ to the left of $c$ .

$a = \frac{11 + c + 2 c}{2}$

$b = - \frac{7}{2} - \frac{c}{2}$

$2 b + 4 c = 9$

Step 4.2.1.9

Add $c$ and $2 c$ .

$a = \frac{11 + 3 c}{2}$

$b = - \frac{7}{2} - \frac{c}{2}$

$2 b + 4 c = 9$

$a = \frac{11 + 3 c}{2}$

$b = - \frac{7}{2} - \frac{c}{2}$

$2 b + 4 c = 9$

$a = \frac{11 + 3 c}{2}$

$b = - \frac{7}{2} - \frac{c}{2}$

$2 b + 4 c = 9$

Step 4.3

Replace all occurrences of $b$ in $2 b + 4 c = 9$ with $- \frac{7}{2} - \frac{c}{2}$ .

$2 (- \frac{7}{2} - \frac{c}{2}) + 4 c = 9$

$a = \frac{11 + 3 c}{2}$

$b = - \frac{7}{2} - \frac{c}{2}$

Step 4.4

Simplify the left side.

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Step 4.4.1

Simplify $2 (- \frac{7}{2} - \frac{c}{2}) + 4 c$ .

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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.

$2 (- \frac{7}{2}) + 2 (- \frac{c}{2}) + 4 c = 9$

$a = \frac{11 + 3 c}{2}$

$b = - \frac{7}{2} - \frac{c}{2}$

Step 4.4.1.1.2

Cancel the common factor of $2$ .

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Step 4.4.1.1.2.1

Move the leading negative in $- \frac{7}{2}$ into the numerator.

$2 (\frac{- 7}{2}) + 2 (- \frac{c}{2}) + 4 c = 9$

$a = \frac{11 + 3 c}{2}$

$b = - \frac{7}{2} - \frac{c}{2}$

Step 4.4.1.1.2.2

Cancel the common factor.

$2 (\frac{- 7}{2}) + 2 (- \frac{c}{2}) + 4 c = 9$

$a = \frac{11 + 3 c}{2}$

$b = - \frac{7}{2} - \frac{c}{2}$

Step 4.4.1.1.2.3

Rewrite the expression.

$- 7 + 2 (- \frac{c}{2}) + 4 c = 9$

$a = \frac{11 + 3 c}{2}$

$b = - \frac{7}{2} - \frac{c}{2}$

$- 7 + 2 (- \frac{c}{2}) + 4 c = 9$

$a = \frac{11 + 3 c}{2}$

$b = - \frac{7}{2} - \frac{c}{2}$

Step 4.4.1.1.3

Cancel the common factor of $2$ .

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Step 4.4.1.1.3.1

Move the leading negative in $- \frac{c}{2}$ into the numerator.

$- 7 + 2 (\frac{- c}{2}) + 4 c = 9$

$a = \frac{11 + 3 c}{2}$

$b = - \frac{7}{2} - \frac{c}{2}$

Step 4.4.1.1.3.2

Cancel the common factor.

$- 7 + 2 (\frac{- c}{2}) + 4 c = 9$

$a = \frac{11 + 3 c}{2}$

$b = - \frac{7}{2} - \frac{c}{2}$

Step 4.4.1.1.3.3

Rewrite the expression.

$- 7 - c + 4 c = 9$

$a = \frac{11 + 3 c}{2}$

$b = - \frac{7}{2} - \frac{c}{2}$

$- 7 - c + 4 c = 9$

$a = \frac{11 + 3 c}{2}$

$b = - \frac{7}{2} - \frac{c}{2}$

$- 7 - c + 4 c = 9$

$a = \frac{11 + 3 c}{2}$

$b = - \frac{7}{2} - \frac{c}{2}$

Step 4.4.1.2

Add $- c$ and $4 c$ .

$- 7 + 3 c = 9$

$a = \frac{11 + 3 c}{2}$

$b = - \frac{7}{2} - \frac{c}{2}$

$- 7 + 3 c = 9$

$a = \frac{11 + 3 c}{2}$

$b = - \frac{7}{2} - \frac{c}{2}$

$- 7 + 3 c = 9$

$a = \frac{11 + 3 c}{2}$

$b = - \frac{7}{2} - \frac{c}{2}$

$- 7 + 3 c = 9$

$a = \frac{11 + 3 c}{2}$

$b = - \frac{7}{2} - \frac{c}{2}$

Step 5

Solve for $c$ in $- 7 + 3 c = 9$ .

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Step 5.1

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

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Step 5.1.1

Add $7$ to both sides of the equation.

$3 c = 9 + 7$

$a = \frac{11 + 3 c}{2}$

$b = - \frac{7}{2} - \frac{c}{2}$

Step 5.1.2

Add $9$ and $7$ .

$3 c = 16$

$a = \frac{11 + 3 c}{2}$

$b = - \frac{7}{2} - \frac{c}{2}$

$3 c = 16$

$a = \frac{11 + 3 c}{2}$

$b = - \frac{7}{2} - \frac{c}{2}$

Step 5.2

Divide each term in $3 c = 16$ by $3$ and simplify.

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Step 5.2.1

Divide each term in $3 c = 16$ by $3$ .

$\frac{3 c}{3} = \frac{16}{3}$

$a = \frac{11 + 3 c}{2}$

$b = - \frac{7}{2} - \frac{c}{2}$

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Cancel the common factor of $3$ .

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Step 5.2.2.1.1

Cancel the common factor.

$\frac{3 c}{3} = \frac{16}{3}$

$a = \frac{11 + 3 c}{2}$

$b = - \frac{7}{2} - \frac{c}{2}$

Step 5.2.2.1.2

Divide $c$ by $1$ .

$c = \frac{16}{3}$

$a = \frac{11 + 3 c}{2}$

$b = - \frac{7}{2} - \frac{c}{2}$

$c = \frac{16}{3}$

$a = \frac{11 + 3 c}{2}$

$b = - \frac{7}{2} - \frac{c}{2}$

$c = \frac{16}{3}$

$a = \frac{11 + 3 c}{2}$

$b = - \frac{7}{2} - \frac{c}{2}$

$c = \frac{16}{3}$

$a = \frac{11 + 3 c}{2}$

$b = - \frac{7}{2} - \frac{c}{2}$

$c = \frac{16}{3}$

$a = \frac{11 + 3 c}{2}$

$b = - \frac{7}{2} - \frac{c}{2}$

Step 6

Replace all occurrences of $c$ with $\frac{16}{3}$ in each equation.

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Step 6.1

Replace all occurrences of $c$ in $a = \frac{11 + 3 c}{2}$ with $\frac{16}{3}$ .

$a = \frac{11 + 3 (\frac{16}{3})}{2}$

$c = \frac{16}{3}$

$b = - \frac{7}{2} - \frac{c}{2}$

Step 6.2

Simplify the right side.

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Step 6.2.1

Simplify the numerator.

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Step 6.2.1.1

Cancel the common factor of $3$ .

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Step 6.2.1.1.1

Cancel the common factor.

$a = \frac{11 + 3 (\frac{16}{3})}{2}$

$c = \frac{16}{3}$

$b = - \frac{7}{2} - \frac{c}{2}$

Step 6.2.1.1.2

Rewrite the expression.

$a = \frac{11 + 16}{2}$

$c = \frac{16}{3}$

$b = - \frac{7}{2} - \frac{c}{2}$

$a = \frac{11 + 16}{2}$

$c = \frac{16}{3}$

$b = - \frac{7}{2} - \frac{c}{2}$

Step 6.2.1.2

Add $11$ and $16$ .

$a = \frac{27}{2}$

$c = \frac{16}{3}$

$b = - \frac{7}{2} - \frac{c}{2}$

$a = \frac{27}{2}$

$c = \frac{16}{3}$

$b = - \frac{7}{2} - \frac{c}{2}$

$a = \frac{27}{2}$

$c = \frac{16}{3}$

$b = - \frac{7}{2} - \frac{c}{2}$

Step 6.3

Replace all occurrences of $c$ in $b = - \frac{7}{2} - \frac{c}{2}$ with $\frac{16}{3}$ .

$b = - \frac{7}{2} - \frac{\frac{16}{3}}{2}$

$a = \frac{27}{2}$

$c = \frac{16}{3}$

Step 6.4

Simplify the right side.

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Step 6.4.1

Simplify $- \frac{7}{2} - \frac{\frac{16}{3}}{2}$ .

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Step 6.4.1.1

Combine the numerators over the common denominator.

$b = \frac{- 7 - \frac{16}{3}}{2}$

$a = \frac{27}{2}$

$c = \frac{16}{3}$

Step 6.4.1.2

To write $- 7$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$b = \frac{- 7 \cdot \frac{3}{3} - \frac{16}{3}}{2}$

$a = \frac{27}{2}$

$c = \frac{16}{3}$

Step 6.4.1.3

Combine $- 7$ and $\frac{3}{3}$ .

$b = \frac{\frac{- 7 \cdot 3}{3} - \frac{16}{3}}{2}$

$a = \frac{27}{2}$

$c = \frac{16}{3}$

Step 6.4.1.4

Combine the numerators over the common denominator.

$b = \frac{\frac{- 7 \cdot 3 - 16}{3}}{2}$

$a = \frac{27}{2}$

$c = \frac{16}{3}$

Step 6.4.1.5

Simplify the numerator.

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Step 6.4.1.5.1

Multiply $- 7$ by $3$ .

$b = \frac{\frac{- 21 - 16}{3}}{2}$

$a = \frac{27}{2}$

$c = \frac{16}{3}$

Step 6.4.1.5.2

Subtract $16$ from $- 21$ .

$b = \frac{\frac{- 37}{3}}{2}$

$a = \frac{27}{2}$

$c = \frac{16}{3}$

$b = \frac{\frac{- 37}{3}}{2}$

$a = \frac{27}{2}$

$c = \frac{16}{3}$

Step 6.4.1.6

Move the negative in front of the fraction.

$b = \frac{- \frac{37}{3}}{2}$

$a = \frac{27}{2}$

$c = \frac{16}{3}$

Step 6.4.1.7

Multiply the numerator by the reciprocal of the denominator.

$b = - \frac{37}{3} \cdot \frac{1}{2}$

$a = \frac{27}{2}$

$c = \frac{16}{3}$

Step 6.4.1.8

Multiply $- \frac{37}{3} \cdot \frac{1}{2}$ .

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Step 6.4.1.8.1

Multiply $\frac{1}{2}$ by $\frac{37}{3}$ .

$b = - \frac{37}{2 \cdot 3}$

$a = \frac{27}{2}$

$c = \frac{16}{3}$

Step 6.4.1.8.2

Multiply $2$ by $3$ .

$b = - \frac{37}{6}$

$a = \frac{27}{2}$

$c = \frac{16}{3}$

$b = - \frac{37}{6}$

$a = \frac{27}{2}$

$c = \frac{16}{3}$

$b = - \frac{37}{6}$

$a = \frac{27}{2}$

$c = \frac{16}{3}$

$b = - \frac{37}{6}$

$a = \frac{27}{2}$

$c = \frac{16}{3}$

$b = - \frac{37}{6}$

$a = \frac{27}{2}$

$c = \frac{16}{3}$

Step 7

List all of the solutions.

$b = - \frac{37}{6}, a = \frac{27}{2}, c = \frac{16}{3}$