Finite Math Examples

$a x + b y = 0$ , $x + y = 8$

Step 1

Solve the equation for $a$ .

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Simplify each term.

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Apply the distributive property.

$a x + b \cdot 8 + b (- x) = 0$

$y = 8 - x$

Move $8$ to the left of $b$ .

$a x + 8 \cdot b + b (- x) = 0$

$y = 8 - x$

Rewrite using the commutative property of multiplication.

$a x + 8 b - b x = 0$

$y = 8 - x$

$a x + 8 b - b x = 0$

$y = 8 - x$

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

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Subtract $8 b$ from both sides of the equation.

$a x - b x = - 8 b$

$y = 8 - x$

Add $b x$ to both sides of the equation.

$a x = - 8 b + b x$

$y = 8 - x$

$a x = - 8 b + b x$

$y = 8 - x$

Divide each term in $a x = - 8 b + b x$ by $x$ and simplify.

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Divide each term in $a x = - 8 b + b x$ by $x$ .

$\frac{a x}{x} = \frac{- 8 b}{x} + \frac{b x}{x}$

$y = 8 - x$

Simplify the left side.

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Cancel the common factor of $x$ .

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Cancel the common factor.

$\frac{a x}{x} = \frac{- 8 b}{x} + \frac{b x}{x}$

$y = 8 - x$

Divide $a$ by $1$ .

$a = \frac{- 8 b}{x} + \frac{b x}{x}$

$y = 8 - x$

$a = \frac{- 8 b}{x} + \frac{b x}{x}$

$y = 8 - x$

$a = \frac{- 8 b}{x} + \frac{b x}{x}$

$y = 8 - x$

Simplify the right side.

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Simplify each term.

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Move the negative in front of the fraction.

$a = - \frac{(8) b}{x} + \frac{b x}{x}$

$y = 8 - x$

Cancel the common factor of $x$ .

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Cancel the common factor.

$a = - \frac{8 b}{x} + \frac{b x}{x}$

$y = 8 - x$

Divide $b$ by $1$ .

$a = - \frac{8 b}{x} + b$

$y = 8 - x$

$a = - \frac{8 b}{x} + b$

$y = 8 - x$

$a = - \frac{8 b}{x} + b$

$y = 8 - x$

$a = - \frac{8 b}{x} + b$

$y = 8 - x$

$a = - \frac{8 b}{x} + b$

$y = 8 - x$

$a = - \frac{8 b}{x} + b$

$y = 8 - x$

Step 2

Solve the equation for $b$ .

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Simplify $(- \frac{8 b}{x} + b) \cdot x + b (8 - x)$ .

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Simplify each term.

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Apply the distributive property.

$- \frac{8 b}{x} \cdot x + b x + b (8 - x) = 0$

$y = 8 - x$

Cancel the common factor of $x$ .

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Move the leading negative in $- \frac{8 b}{x}$ into the numerator.

$\frac{- 8 b}{x} \cdot x + b x + b (8 - x) = 0$

$y = 8 - x$

Cancel the common factor.

$\frac{- 8 b}{x} \cdot x + b x + b (8 - x) = 0$

$y = 8 - x$

Rewrite the expression.

$- 8 b + b x + b (8 - x) = 0$

$y = 8 - x$

$- 8 b + b x + b (8 - x) = 0$

$y = 8 - x$

Apply the distributive property.

$- 8 b + b x + b \cdot 8 + b (- x) = 0$

$y = 8 - x$

Move $8$ to the left of $b$ .

$- 8 b + b x + 8 \cdot b + b (- x) = 0$

$y = 8 - x$

Rewrite using the commutative property of multiplication.

$- 8 b + b x + 8 b - b x = 0$

$y = 8 - x$

$- 8 b + b x + 8 b - b x = 0$

$y = 8 - x$

Combine the opposite terms in $- 8 b + b x + 8 b - b x$ .

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Add $- 8 b$ and $8 b$ .

$b x + 0 - b x = 0$

$y = 8 - x$

Add $b x$ and $0$ .

$b x - b x = 0$

$y = 8 - x$

Subtract $b x$ from $b x$ .

$0 = 0$

$y = 8 - x$

$0 = 0$

$y = 8 - x$

$0 = 0$

$y = 8 - x$

Since $0 = 0$ , the equation will always be true.

Always true

$y = 8 - x$

Always true

$y = 8 - x$

Step 3

Simplify the right side.

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Reorder $8$ and $- x$ .

$y = - x + 8$

Always true

$y = - x + 8$

Always true