Finite Math Examples

$f = 2 x + 30 y$ , $2 x + 10 y = 96$ , $1 x + 10 y = 90$

Step 1

Subtract $10 y$ from both sides of the equation.

$x = 90 - 10 y$

$f = 2 x + 30 y$

$2 x + 10 y = 96$

Step 2

Replace all occurrences of $x$ with $90 - 10 y$ in each equation.

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Step 2.1

Replace all occurrences of $x$ in $f = 2 x + 30 y$ with $90 - 10 y$ .

$f = 2 (90 - 10 y) + 30 y$

$x = 90 - 10 y$

$2 x + 10 y = 96$

Step 2.2

Simplify the right side.

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Step 2.2.1

Simplify $2 (90 - 10 y) + 30 y$ .

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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.

$f = 2 \cdot 90 + 2 (- 10 y) + 30 y$

$x = 90 - 10 y$

$2 x + 10 y = 96$

Step 2.2.1.1.2

Multiply $2$ by $90$ .

$f = 180 + 2 (- 10 y) + 30 y$

$x = 90 - 10 y$

$2 x + 10 y = 96$

Step 2.2.1.1.3

Multiply $- 10$ by $2$ .

$f = 180 - 20 y + 30 y$

$x = 90 - 10 y$

$2 x + 10 y = 96$

$f = 180 - 20 y + 30 y$

$x = 90 - 10 y$

$2 x + 10 y = 96$

Step 2.2.1.2

Add $- 20 y$ and $30 y$ .

$f = 180 + 10 y$

$x = 90 - 10 y$

$2 x + 10 y = 96$

$f = 180 + 10 y$

$x = 90 - 10 y$

$2 x + 10 y = 96$

$f = 180 + 10 y$

$x = 90 - 10 y$

$2 x + 10 y = 96$

Step 2.3

Replace all occurrences of $x$ in $2 x + 10 y = 96$ with $90 - 10 y$ .

$2 (90 - 10 y) + 10 y = 96$

$f = 180 + 10 y$

$x = 90 - 10 y$

Step 2.4

Simplify the left side.

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Step 2.4.1

Simplify $2 (90 - 10 y) + 10 y$ .

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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.

$2 \cdot 90 + 2 (- 10 y) + 10 y = 96$

$f = 180 + 10 y$

$x = 90 - 10 y$

Step 2.4.1.1.2

Multiply $2$ by $90$ .

$180 + 2 (- 10 y) + 10 y = 96$

$f = 180 + 10 y$

$x = 90 - 10 y$

Step 2.4.1.1.3

Multiply $- 10$ by $2$ .

$180 - 20 y + 10 y = 96$

$f = 180 + 10 y$

$x = 90 - 10 y$

$180 - 20 y + 10 y = 96$

$f = 180 + 10 y$

$x = 90 - 10 y$

Step 2.4.1.2

Add $- 20 y$ and $10 y$ .

$180 - 10 y = 96$

$f = 180 + 10 y$

$x = 90 - 10 y$

$180 - 10 y = 96$

$f = 180 + 10 y$

$x = 90 - 10 y$

$180 - 10 y = 96$

$f = 180 + 10 y$

$x = 90 - 10 y$

$180 - 10 y = 96$

$f = 180 + 10 y$

$x = 90 - 10 y$

Step 3

Solve for $y$ in $180 - 10 y = 96$ .

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Step 3.1

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

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Step 3.1.1

Subtract $180$ from both sides of the equation.

$- 10 y = 96 - 180$

$f = 180 + 10 y$

$x = 90 - 10 y$

Step 3.1.2

Subtract $180$ from $96$ .

$- 10 y = - 84$

$f = 180 + 10 y$

$x = 90 - 10 y$

$- 10 y = - 84$

$f = 180 + 10 y$

$x = 90 - 10 y$

Step 3.2

Divide each term in $- 10 y = - 84$ by $- 10$ and simplify.

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Step 3.2.1

Divide each term in $- 10 y = - 84$ by $- 10$ .

$\frac{- 10 y}{- 10} = \frac{- 84}{- 10}$

$f = 180 + 10 y$

$x = 90 - 10 y$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $- 10$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{- 10 y}{- 10} = \frac{- 84}{- 10}$

$f = 180 + 10 y$

$x = 90 - 10 y$

Step 3.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 84}{- 10}$

$f = 180 + 10 y$

$x = 90 - 10 y$

$y = \frac{- 84}{- 10}$

$f = 180 + 10 y$

$x = 90 - 10 y$

$y = \frac{- 84}{- 10}$

$f = 180 + 10 y$

$x = 90 - 10 y$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Cancel the common factor of $- 84$ and $- 10$ .

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Step 3.2.3.1.1

Factor $- 2$ out of $- 84$ .

$y = \frac{- 2 \cdot 42}{- 10}$

$f = 180 + 10 y$

$x = 90 - 10 y$

Step 3.2.3.1.2

Cancel the common factors.

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Step 3.2.3.1.2.1

Factor $- 2$ out of $- 10$ .

$y = \frac{- 2 \cdot 42}{- 2 \cdot 5}$

$f = 180 + 10 y$

$x = 90 - 10 y$

Step 3.2.3.1.2.2

Cancel the common factor.

$y = \frac{- 2 \cdot 42}{- 2 \cdot 5}$

$f = 180 + 10 y$

$x = 90 - 10 y$

Step 3.2.3.1.2.3

Rewrite the expression.

$y = \frac{42}{5}$

$f = 180 + 10 y$

$x = 90 - 10 y$

$y = \frac{42}{5}$

$f = 180 + 10 y$

$x = 90 - 10 y$

$y = \frac{42}{5}$

$f = 180 + 10 y$

$x = 90 - 10 y$

$y = \frac{42}{5}$

$f = 180 + 10 y$

$x = 90 - 10 y$

$y = \frac{42}{5}$

$f = 180 + 10 y$

$x = 90 - 10 y$

$y = \frac{42}{5}$

$f = 180 + 10 y$

$x = 90 - 10 y$

Step 4

Replace all occurrences of $y$ with $\frac{42}{5}$ in each equation.

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Step 4.1

Replace all occurrences of $y$ in $f = 180 + 10 y$ with $\frac{42}{5}$ .

$f = 180 + 10 (\frac{42}{5})$

$y = \frac{42}{5}$

$x = 90 - 10 y$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $180 + 10 (\frac{42}{5})$ .

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Step 4.2.1.1

Simplify each term.

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Step 4.2.1.1.1

Cancel the common factor of $5$ .

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Step 4.2.1.1.1.1

Factor $5$ out of $10$ .

$f = 180 + 5 (2) (\frac{42}{5})$

$y = \frac{42}{5}$

$x = 90 - 10 y$

Step 4.2.1.1.1.2

Cancel the common factor.

$f = 180 + 5 \cdot (2 (\frac{42}{5}))$

$y = \frac{42}{5}$

$x = 90 - 10 y$

Step 4.2.1.1.1.3

Rewrite the expression.

$f = 180 + 2 \cdot 42$

$y = \frac{42}{5}$

$x = 90 - 10 y$

$f = 180 + 2 \cdot 42$

$y = \frac{42}{5}$

$x = 90 - 10 y$

Step 4.2.1.1.2

Multiply $2$ by $42$ .

$f = 180 + 84$

$y = \frac{42}{5}$

$x = 90 - 10 y$

$f = 180 + 84$

$y = \frac{42}{5}$

$x = 90 - 10 y$

Step 4.2.1.2

Add $180$ and $84$ .

$f = 264$

$y = \frac{42}{5}$

$x = 90 - 10 y$

$f = 264$

$y = \frac{42}{5}$

$x = 90 - 10 y$

$f = 264$

$y = \frac{42}{5}$

$x = 90 - 10 y$

Step 4.3

Replace all occurrences of $y$ in $x = 90 - 10 y$ with $\frac{42}{5}$ .

$x = 90 - 10 (\frac{42}{5})$

$f = 264$

$y = \frac{42}{5}$

Step 4.4

Simplify the right side.

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Step 4.4.1

Simplify $90 - 10 (\frac{42}{5})$ .

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Step 4.4.1.1

Simplify each term.

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Step 4.4.1.1.1

Cancel the common factor of $5$ .

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Step 4.4.1.1.1.1

Factor $5$ out of $- 10$ .

$x = 90 + 5 (- 2) (\frac{42}{5})$

$f = 264$

$y = \frac{42}{5}$

Step 4.4.1.1.1.2

Cancel the common factor.

$x = 90 + 5 \cdot (- 2 (\frac{42}{5}))$

$f = 264$

$y = \frac{42}{5}$

Step 4.4.1.1.1.3

Rewrite the expression.

$x = 90 - 2 \cdot 42$

$f = 264$

$y = \frac{42}{5}$

$x = 90 - 2 \cdot 42$

$f = 264$

$y = \frac{42}{5}$

Step 4.4.1.1.2

Multiply $- 2$ by $42$ .

$x = 90 - 84$

$f = 264$

$y = \frac{42}{5}$

$x = 90 - 84$

$f = 264$

$y = \frac{42}{5}$

Step 4.4.1.2

Subtract $84$ from $90$ .

$x = 6$

$f = 264$

$y = \frac{42}{5}$

$x = 6$

$f = 264$

$y = \frac{42}{5}$

$x = 6$

$f = 264$

$y = \frac{42}{5}$

$x = 6$

$f = 264$

$y = \frac{42}{5}$

Step 5

List all of the solutions.

$x = 6, f = 264, y = \frac{42}{5}$