Finite Math Examples

$144 = a {(1)}^{2} + b (1) + 128$ , $0 = a {(4)}^{2} + b (4) + 128$

Step 1

Solve for $a$ in $144 = a + b + 128$ .

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Step 1.1

Rewrite the equation as $a + b + 128 = 144$ .

$a + b + 128 = 144$

$0 = a \cdot 4^{2} + b (4) + 128$

Step 1.2

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

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Step 1.2.1

Subtract $b$ from both sides of the equation.

$a + 128 = 144 - b$

$0 = a \cdot 4^{2} + b (4) + 128$

Step 1.2.2

Subtract $128$ from both sides of the equation.

$a = 144 - b - 128$

$0 = a \cdot 4^{2} + b (4) + 128$

Step 1.2.3

Subtract $128$ from $144$ .

$a = - b + 16$

$0 = a \cdot 4^{2} + b (4) + 128$

$a = - b + 16$

$0 = a \cdot 4^{2} + b (4) + 128$

$a = - b + 16$

$0 = a \cdot 4^{2} + b (4) + 128$

Step 2

Replace all occurrences of $a$ with $- b + 16$ in each equation.

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Step 2.1

Replace all occurrences of $a$ in $0 = a \cdot 4^{2} + b (4) + 128$ with $- b + 16$ .

$0 = (- b + 16) \cdot 4^{2} + b (4) + 128$

$a = - b + 16$

Step 2.2

Simplify the right side.

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Step 2.2.1

Simplify $(- b + 16) \cdot 4^{2} + b (4) + 128$ .

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Step 2.2.1.1

Simplify each term.

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Step 2.2.1.1.1

Raise $4$ to the power of $2$ .

$0 = (- b + 16) \cdot 16 + b (4) + 128$

$a = - b + 16$

Step 2.2.1.1.2

Apply the distributive property.

$0 = - b \cdot 16 + 16 \cdot 16 + b (4) + 128$

$a = - b + 16$

Step 2.2.1.1.3

Multiply $16$ by $- 1$ .

$0 = - 16 b + 16 \cdot 16 + b (4) + 128$

$a = - b + 16$

Step 2.2.1.1.4

Multiply $16$ by $16$ .

$0 = - 16 b + 256 + b (4) + 128$

$a = - b + 16$

Step 2.2.1.1.5

Move $4$ to the left of $b$ .

$0 = - 16 b + 256 + 4 b + 128$

$a = - b + 16$

$0 = - 16 b + 256 + 4 b + 128$

$a = - b + 16$

Step 2.2.1.2

Simplify by adding terms.

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Step 2.2.1.2.1

Add $- 16 b$ and $4 b$ .

$0 = - 12 b + 256 + 128$

$a = - b + 16$

Step 2.2.1.2.2

Add $256$ and $128$ .

$0 = - 12 b + 384$

$a = - b + 16$

$0 = - 12 b + 384$

$a = - b + 16$

$0 = - 12 b + 384$

$a = - b + 16$

$0 = - 12 b + 384$

$a = - b + 16$

$0 = - 12 b + 384$

$a = - b + 16$

Step 3

Solve for $b$ in $0 = - 12 b + 384$ .

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Step 3.1

Rewrite the equation as $- 12 b + 384 = 0$ .

$- 12 b + 384 = 0$

$a = - b + 16$

Step 3.2

Subtract $384$ from both sides of the equation.

$- 12 b = - 384$

$a = - b + 16$

Step 3.3

Divide each term in $- 12 b = - 384$ by $- 12$ and simplify.

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Step 3.3.1

Divide each term in $- 12 b = - 384$ by $- 12$ .

$\frac{- 12 b}{- 12} = \frac{- 384}{- 12}$

$a = - b + 16$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $- 12$ .

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Step 3.3.2.1.1

Cancel the common factor.

$\frac{- 12 b}{- 12} = \frac{- 384}{- 12}$

$a = - b + 16$

Step 3.3.2.1.2

Divide $b$ by $1$ .

$b = \frac{- 384}{- 12}$

$a = - b + 16$

$b = \frac{- 384}{- 12}$

$a = - b + 16$

$b = \frac{- 384}{- 12}$

$a = - b + 16$

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

Divide $- 384$ by $- 12$ .

$b = 32$

$a = - b + 16$

$b = 32$

$a = - b + 16$

$b = 32$

$a = - b + 16$

$b = 32$

$a = - b + 16$

Step 4

Replace all occurrences of $b$ with $32$ in each equation.

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Step 4.1

Replace all occurrences of $b$ in $a = - b + 16$ with $32$ .

$a = - (32) + 16$

$b = 32$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $- (32) + 16$ .

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Step 4.2.1.1

Multiply $- 1$ by $32$ .

$a = - 32 + 16$

$b = 32$

Step 4.2.1.2

Add $- 32$ and $16$ .

$a = - 16$

$b = 32$

$a = - 16$

$b = 32$

$a = - 16$

$b = 32$

$a = - 16$

$b = 32$

Step 5

List all of the solutions.

$a = - 16, b = 32$