Finite Math Examples

$a = 3 b - 1$ , $b = 2 a + 2$

Step 1

Replace all occurrences of $a$ with $3 b - 1$ in each equation.

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Step 1.1

Replace all occurrences of $a$ in $b = 2 a + 2$ with $3 b - 1$ .

$b = 2 (3 b - 1) + 2$

$a = 3 b - 1$

Step 1.2

Simplify the right side.

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Step 1.2.1

Simplify $2 (3 b - 1) + 2$ .

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Step 1.2.1.1

Simplify each term.

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Step 1.2.1.1.1

Apply the distributive property.

$b = 2 (3 b) + 2 \cdot - 1 + 2$

$a = 3 b - 1$

Step 1.2.1.1.2

Multiply $3$ by $2$ .

$b = 6 b + 2 \cdot - 1 + 2$

$a = 3 b - 1$

Step 1.2.1.1.3

Multiply $2$ by $- 1$ .

$b = 6 b - 2 + 2$

$a = 3 b - 1$

$b = 6 b - 2 + 2$

$a = 3 b - 1$

Step 1.2.1.2

Combine the opposite terms in $6 b - 2 + 2$ .

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Step 1.2.1.2.1

Add $- 2$ and $2$ .

$b = 6 b + 0$

$a = 3 b - 1$

Step 1.2.1.2.2

Add $6 b$ and $0$ .

$b = 6 b$

$a = 3 b - 1$

$b = 6 b$

$a = 3 b - 1$

$b = 6 b$

$a = 3 b - 1$

$b = 6 b$

$a = 3 b - 1$

$b = 6 b$

$a = 3 b - 1$

Step 2

Solve for $b$ in $b = 6 b$ .

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Step 2.1

Move all terms containing $b$ to the left side of the equation.

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Step 2.1.1

Subtract $6 b$ from both sides of the equation.

$b - 6 b = 0$

$a = 3 b - 1$

Step 2.1.2

Subtract $6 b$ from $b$ .

$- 5 b = 0$

$a = 3 b - 1$

$- 5 b = 0$

$a = 3 b - 1$

Step 2.2

Divide each term in $- 5 b = 0$ by $- 5$ and simplify.

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Step 2.2.1

Divide each term in $- 5 b = 0$ by $- 5$ .

$\frac{- 5 b}{- 5} = \frac{0}{- 5}$

$a = 3 b - 1$

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Cancel the common factor of $- 5$ .

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Step 2.2.2.1.1

Cancel the common factor.

$\frac{- 5 b}{- 5} = \frac{0}{- 5}$

$a = 3 b - 1$

Step 2.2.2.1.2

Divide $b$ by $1$ .

$b = \frac{0}{- 5}$

$a = 3 b - 1$

$b = \frac{0}{- 5}$

$a = 3 b - 1$

$b = \frac{0}{- 5}$

$a = 3 b - 1$

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Divide $0$ by $- 5$ .

$b = 0$

$a = 3 b - 1$

$b = 0$

$a = 3 b - 1$

$b = 0$

$a = 3 b - 1$

$b = 0$

$a = 3 b - 1$

Step 3

Replace all occurrences of $b$ with $0$ in each equation.

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Step 3.1

Replace all occurrences of $b$ in $a = 3 b - 1$ with $0$ .

$a = 3 (0) - 1$

$b = 0$

Step 3.2

Simplify the right side.

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Step 3.2.1

Simplify $3 (0) - 1$ .

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Step 3.2.1.1

Multiply $3$ by $0$ .

$a = 0 - 1$

$b = 0$

Step 3.2.1.2

Subtract $1$ from $0$ .

$a = - 1$

$b = 0$

$a = - 1$

$b = 0$

$a = - 1$

$b = 0$

$a = - 1$

$b = 0$

Step 4

List all of the solutions.

$a = - 1, b = 0$