Finite Math Examples

$- a - b = - 2$ , $15 a + 7 b - 9 c = 2$ , $a - c = - 3$

Step 1

Add $c$ to both sides of the equation.

$a = - 3 + c$

$- a - b = - 2$

$15 a + 7 b - 9 c = 2$

Step 2

Replace all occurrences of $a$ with $- 3 + c$ in each equation.

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Step 2.1

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

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

$a = - 3 + c$

$15 a + 7 b - 9 c = 2$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify each term.

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Step 2.2.1.1

Apply the distributive property.

$3 - c - b = - 2$

$a = - 3 + c$

$15 a + 7 b - 9 c = 2$

Step 2.2.1.2

Multiply $- 1$ by $- 3$ .

$3 - c - b = - 2$

$a = - 3 + c$

$15 a + 7 b - 9 c = 2$

$3 - c - b = - 2$

$a = - 3 + c$

$15 a + 7 b - 9 c = 2$

$3 - c - b = - 2$

$a = - 3 + c$

$15 a + 7 b - 9 c = 2$

Step 2.3

Replace all occurrences of $a$ in $15 a + 7 b - 9 c = 2$ with $- 3 + c$ .

$15 (- 3 + c) + 7 b - 9 c = 2$

$3 - c - b = - 2$

$a = - 3 + c$

Step 2.4

Simplify the left side.

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Step 2.4.1

Simplify $15 (- 3 + c) + 7 b - 9 c$ .

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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.

$15 \cdot - 3 + 15 c + 7 b - 9 c = 2$

$3 - c - b = - 2$

$a = - 3 + c$

Step 2.4.1.1.2

Multiply $15$ by $- 3$ .

$- 45 + 15 c + 7 b - 9 c = 2$

$3 - c - b = - 2$

$a = - 3 + c$

$- 45 + 15 c + 7 b - 9 c = 2$

$3 - c - b = - 2$

$a = - 3 + c$

Step 2.4.1.2

Subtract $9 c$ from $15 c$ .

$- 45 + 7 b + 6 c = 2$

$3 - c - b = - 2$

$a = - 3 + c$

$- 45 + 7 b + 6 c = 2$

$3 - c - b = - 2$

$a = - 3 + c$

$- 45 + 7 b + 6 c = 2$

$3 - c - b = - 2$

$a = - 3 + c$

$- 45 + 7 b + 6 c = 2$

$3 - c - b = - 2$

$a = - 3 + c$

Step 3

Reorder $- 3$ and $c$ .

$a = c - 3$

$- 45 + 7 b + 6 c = 2$

$3 - c - b = - 2$

Step 4

Solve for $c$ in $a = c - 3$ .

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Step 4.1

Rewrite the equation as $c - 3 = a$ .

$c - 3 = a$

$- 45 + 7 b + 6 c = 2$

$3 - c - b = - 2$

Step 4.2

Add $3$ to both sides of the equation.

$c = a + 3$

$- 45 + 7 b + 6 c = 2$

$3 - c - b = - 2$

$c = a + 3$

$- 45 + 7 b + 6 c = 2$

$3 - c - b = - 2$

Step 5

Replace all occurrences of $c$ with $a + 3$ in each equation.

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Step 5.1

Replace all occurrences of $c$ in $- 45 + 7 b + 6 c = 2$ with $a + 3$ .

$- 45 + 7 b + 6 (a + 3) = 2$

$c = a + 3$

$3 - c - b = - 2$

Step 5.2

Simplify the left side.

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Step 5.2.1

Simplify $- 45 + 7 b + 6 (a + 3)$ .

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Step 5.2.1.1

Simplify each term.

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Step 5.2.1.1.1

Apply the distributive property.

$- 45 + 7 b + 6 a + 6 \cdot 3 = 2$

$c = a + 3$

$3 - c - b = - 2$

Step 5.2.1.1.2

Multiply $6$ by $3$ .

$- 45 + 7 b + 6 a + 18 = 2$

$c = a + 3$

$3 - c - b = - 2$

$- 45 + 7 b + 6 a + 18 = 2$

$c = a + 3$

$3 - c - b = - 2$

Step 5.2.1.2

Add $- 45$ and $18$ .

$7 b + 6 a - 27 = 2$

$c = a + 3$

$3 - c - b = - 2$

$7 b + 6 a - 27 = 2$

$c = a + 3$

$3 - c - b = - 2$

$7 b + 6 a - 27 = 2$

$c = a + 3$

$3 - c - b = - 2$

Step 5.3

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

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

$7 b + 6 a - 27 = 2$

$c = a + 3$

Step 5.4

Simplify the left side.

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Step 5.4.1

Simplify $3 - (a + 3) - b$ .

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Step 5.4.1.1

Simplify each term.

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Step 5.4.1.1.1

Apply the distributive property.

$3 - a - 1 \cdot 3 - b = - 2$

$7 b + 6 a - 27 = 2$

$c = a + 3$

Step 5.4.1.1.2

Multiply $- 1$ by $3$ .

$3 - a - 3 - b = - 2$

$7 b + 6 a - 27 = 2$

$c = a + 3$

$3 - a - 3 - b = - 2$

$7 b + 6 a - 27 = 2$

$c = a + 3$

Step 5.4.1.2

Combine the opposite terms in $3 - a - 3 - b$ .

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Step 5.4.1.2.1

Subtract $3$ from $3$ .

$- a + 0 - b = - 2$

$7 b + 6 a - 27 = 2$

$c = a + 3$

Step 5.4.1.2.2

Add $- a$ and $0$ .

$- a - b = - 2$

$7 b + 6 a - 27 = 2$

$c = a + 3$

$- a - b = - 2$

$7 b + 6 a - 27 = 2$

$c = a + 3$

$- a - b = - 2$

$7 b + 6 a - 27 = 2$

$c = a + 3$

$- a - b = - 2$

$7 b + 6 a - 27 = 2$

$c = a + 3$

$- a - b = - 2$

$7 b + 6 a - 27 = 2$

$c = a + 3$

Step 6

Solve for $a$ in $- a - b = - 2$ .

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Step 6.1

Add $b$ to both sides of the equation.

$- a = - 2 + b$

$7 b + 6 a - 27 = 2$

$c = a + 3$

Step 6.2

Divide each term in $- a = - 2 + b$ by $- 1$ and simplify.

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Step 6.2.1

Divide each term in $- a = - 2 + b$ by $- 1$ .

$\frac{- a}{- 1} = \frac{- 2}{- 1} + \frac{b}{- 1}$

$7 b + 6 a - 27 = 2$

$c = a + 3$

Step 6.2.2

Simplify the left side.

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Step 6.2.2.1

Dividing two negative values results in a positive value.

$\frac{a}{1} = \frac{- 2}{- 1} + \frac{b}{- 1}$

$7 b + 6 a - 27 = 2$

$c = a + 3$

Step 6.2.2.2

Divide $a$ by $1$ .

$a = \frac{- 2}{- 1} + \frac{b}{- 1}$

$7 b + 6 a - 27 = 2$

$c = a + 3$

$a = \frac{- 2}{- 1} + \frac{b}{- 1}$

$7 b + 6 a - 27 = 2$

$c = a + 3$

Step 6.2.3

Simplify the right side.

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Step 6.2.3.1

Simplify each term.

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Step 6.2.3.1.1

Divide $- 2$ by $- 1$ .

$a = 2 + \frac{b}{- 1}$

$7 b + 6 a - 27 = 2$

$c = a + 3$

Step 6.2.3.1.2

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

$a = 2 - 1 \cdot b$

$7 b + 6 a - 27 = 2$

$c = a + 3$

Step 6.2.3.1.3

Rewrite $- 1 \cdot b$ as $- b$ .

$a = 2 - b$

$7 b + 6 a - 27 = 2$

$c = a + 3$

$a = 2 - b$

$7 b + 6 a - 27 = 2$

$c = a + 3$

$a = 2 - b$

$7 b + 6 a - 27 = 2$

$c = a + 3$

$a = 2 - b$

$7 b + 6 a - 27 = 2$

$c = a + 3$

$a = 2 - b$

$7 b + 6 a - 27 = 2$

$c = a + 3$

Step 7

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

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Step 7.1

Replace all occurrences of $a$ in $7 b + 6 a - 27 = 2$ with $2 - b$ .

$7 b + 6 (2 - b) - 27 = 2$

$a = 2 - b$

$c = a + 3$

Step 7.2

Simplify the left side.

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Step 7.2.1

Simplify $7 b + 6 (2 - b) - 27$ .

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Step 7.2.1.1

Simplify each term.

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Step 7.2.1.1.1

Apply the distributive property.

$7 b + 6 \cdot 2 + 6 (- b) - 27 = 2$

$a = 2 - b$

$c = a + 3$

Step 7.2.1.1.2

Multiply $6$ by $2$ .

$7 b + 12 + 6 (- b) - 27 = 2$

$a = 2 - b$

$c = a + 3$

Step 7.2.1.1.3

Multiply $- 1$ by $6$ .

$7 b + 12 - 6 b - 27 = 2$

$a = 2 - b$

$c = a + 3$

$7 b + 12 - 6 b - 27 = 2$

$a = 2 - b$

$c = a + 3$

Step 7.2.1.2

Simplify by adding terms.

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Step 7.2.1.2.1

Subtract $6 b$ from $7 b$ .

$b + 12 - 27 = 2$

$a = 2 - b$

$c = a + 3$

Step 7.2.1.2.2

Subtract $27$ from $12$ .

$b - 15 = 2$

$a = 2 - b$

$c = a + 3$

$b - 15 = 2$

$a = 2 - b$

$c = a + 3$

$b - 15 = 2$

$a = 2 - b$

$c = a + 3$

$b - 15 = 2$

$a = 2 - b$

$c = a + 3$

Step 7.3

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

$c = (2 - b) + 3$

$b - 15 = 2$

$a = 2 - b$

Step 7.4

Simplify the right side.

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Step 7.4.1

Add $2$ and $3$ .

$c = - b + 5$

$b - 15 = 2$

$a = 2 - b$

$c = - b + 5$

$b - 15 = 2$

$a = 2 - b$

$c = - b + 5$

$b - 15 = 2$

$a = 2 - b$

Step 8

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

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Step 8.1

Add $15$ to both sides of the equation.

$b = 2 + 15$

$c = - b + 5$

$a = 2 - b$

Step 8.2

Add $2$ and $15$ .

$b = 17$

$c = - b + 5$

$a = 2 - b$

$b = 17$

$c = - b + 5$

$a = 2 - b$

Step 9

Replace all occurrences of $b$ with $17$ in each equation.

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Step 9.1

Replace all occurrences of $b$ in $c = - b + 5$ with $17$ .

$c = - (17) + 5$

$b = 17$

$a = 2 - b$

Step 9.2

Simplify the right side.

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Step 9.2.1

Simplify $- (17) + 5$ .

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Step 9.2.1.1

Multiply $- 1$ by $17$ .

$c = - 17 + 5$

$b = 17$

$a = 2 - b$

Step 9.2.1.2

Add $- 17$ and $5$ .

$c = - 12$

$b = 17$

$a = 2 - b$

$c = - 12$

$b = 17$

$a = 2 - b$

$c = - 12$

$b = 17$

$a = 2 - b$

Step 9.3

Replace all occurrences of $b$ in $a = 2 - b$ with $17$ .

$a = 2 - (17)$

$c = - 12$

$b = 17$

Step 9.4

Simplify the right side.

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Step 9.4.1

Simplify $2 - (17)$ .

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Step 9.4.1.1

Multiply $- 1$ by $17$ .

$a = 2 - 17$

$c = - 12$

$b = 17$

Step 9.4.1.2

Subtract $17$ from $2$ .

$a = - 15$

$c = - 12$

$b = 17$

$a = - 15$

$c = - 12$

$b = 17$

$a = - 15$

$c = - 12$

$b = 17$

$a = - 15$

$c = - 12$

$b = 17$

Step 10

List all of the solutions.

$a = - 15, c = - 12, b = 17$