Finite Math Examples

$\frac{x}{6} - \frac{y}{4} = 1$ , $\frac{x}{4} + y = - 4$

Step 1

Subtract $\frac{x}{4}$ from both sides of the equation.

$y = - 4 - \frac{x}{4}$

$\frac{x}{6} - \frac{y}{4} = 1$

Step 2

Replace all occurrences of $y$ with $- 4 - \frac{x}{4}$ in each equation.

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Step 2.1

Replace all occurrences of $y$ in $\frac{x}{6} - \frac{y}{4} = 1$ with $- 4 - \frac{x}{4}$ .

$\frac{x}{6} - \frac{- 4 - \frac{x}{4}}{4} = 1$

$y = - 4 - \frac{x}{4}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $\frac{x}{6} - \frac{- 4 - \frac{x}{4}}{4}$ .

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Step 2.2.1.1

Simplify each term.

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Step 2.2.1.1.1

Simplify the numerator.

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Step 2.2.1.1.1.1

Factor $- 1$ out of $- 4 - \frac{x}{4}$ .

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Step 2.2.1.1.1.1.1

Reorder $- 4$ and $- \frac{x}{4}$ .

$\frac{x}{6} - \frac{- \frac{x}{4} - 4}{4} = 1$

$y = - 4 - \frac{x}{4}$

Step 2.2.1.1.1.1.2

Factor $- 1$ out of $- \frac{x}{4}$ .

$\frac{x}{6} - \frac{- (\frac{x}{4}) - 4}{4} = 1$

$y = - 4 - \frac{x}{4}$

Step 2.2.1.1.1.1.3

Rewrite $- 4$ as $- 1 (4)$ .

$\frac{x}{6} - \frac{- (\frac{x}{4}) - 1 \cdot 4}{4} = 1$

$y = - 4 - \frac{x}{4}$

Step 2.2.1.1.1.1.4

Factor $- 1$ out of $- (\frac{x}{4}) - 1 (4)$ .

$\frac{x}{6} - \frac{- (\frac{x}{4} + 4)}{4} = 1$

$y = - 4 - \frac{x}{4}$

$\frac{x}{6} - \frac{- (\frac{x}{4} + 4)}{4} = 1$

$y = - 4 - \frac{x}{4}$

Step 2.2.1.1.1.2

To write $4$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{x}{6} - \frac{- (\frac{x}{4} + 4 \cdot \frac{4}{4})}{4} = 1$

$y = - 4 - \frac{x}{4}$

Step 2.2.1.1.1.3

Combine $4$ and $\frac{4}{4}$ .

$\frac{x}{6} - \frac{- (\frac{x}{4} + \frac{4 \cdot 4}{4})}{4} = 1$

$y = - 4 - \frac{x}{4}$

Step 2.2.1.1.1.4

Combine the numerators over the common denominator.

$\frac{x}{6} - \frac{- \frac{x + 4 \cdot 4}{4}}{4} = 1$

$y = - 4 - \frac{x}{4}$

Step 2.2.1.1.1.5

Multiply $4$ by $4$ .

$\frac{x}{6} - \frac{- \frac{x + 16}{4}}{4} = 1$

$y = - 4 - \frac{x}{4}$

$\frac{x}{6} - \frac{- \frac{x + 16}{4}}{4} = 1$

$y = - 4 - \frac{x}{4}$

Step 2.2.1.1.2

Multiply the numerator by the reciprocal of the denominator.

$\frac{x}{6} - (- \frac{x + 16}{4} \cdot \frac{1}{4}) = 1$

$y = - 4 - \frac{x}{4}$

Step 2.2.1.1.3

Multiply $- \frac{x + 16}{4} \cdot \frac{1}{4}$ .

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Step 2.2.1.1.3.1

Multiply $\frac{1}{4}$ by $\frac{x + 16}{4}$ .

$\frac{x}{6} + \frac{x + 16}{4 \cdot 4} = 1$

$y = - 4 - \frac{x}{4}$

Step 2.2.1.1.3.2

Multiply $4$ by $4$ .

$\frac{x}{6} + \frac{x + 16}{16} = 1$

$y = - 4 - \frac{x}{4}$

$\frac{x}{6} + \frac{x + 16}{16} = 1$

$y = - 4 - \frac{x}{4}$

Step 2.2.1.1.4

Multiply $- - \frac{x + 16}{16}$ .

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Step 2.2.1.1.4.1

Multiply $- 1$ by $- 1$ .

$\frac{x}{6} + 1 (\frac{x + 16}{16}) = 1$

$y = - 4 - \frac{x}{4}$

Step 2.2.1.1.4.2

Multiply $\frac{x + 16}{16}$ by $1$ .

$\frac{x}{6} + \frac{x + 16}{16} = 1$

$y = - 4 - \frac{x}{4}$

$\frac{x}{6} + \frac{x + 16}{16} = 1$

$y = - 4 - \frac{x}{4}$

$\frac{x}{6} + \frac{x + 16}{16} = 1$

$y = - 4 - \frac{x}{4}$

Step 2.2.1.2

To write $\frac{x}{6}$ as a fraction with a common denominator, multiply by $\frac{8}{8}$ .

$\frac{x}{6} \cdot \frac{8}{8} + \frac{x + 16}{16} = 1$

$y = - 4 - \frac{x}{4}$

Step 2.2.1.3

To write $\frac{x + 16}{16}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{x}{6} \cdot \frac{8}{8} + \frac{x + 16}{16} \cdot \frac{3}{3} = 1$

$y = - 4 - \frac{x}{4}$

Step 2.2.1.4

Write each expression with a common denominator of $48$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.2.1.4.1

Multiply $\frac{x}{6}$ by $\frac{8}{8}$ .

$\frac{x \cdot 8}{6 \cdot 8} + \frac{x + 16}{16} \cdot \frac{3}{3} = 1$

$y = - 4 - \frac{x}{4}$

Step 2.2.1.4.2

Multiply $6$ by $8$ .

$\frac{x \cdot 8}{48} + \frac{x + 16}{16} \cdot \frac{3}{3} = 1$

$y = - 4 - \frac{x}{4}$

Step 2.2.1.4.3

Multiply $\frac{x + 16}{16}$ by $\frac{3}{3}$ .

$\frac{x \cdot 8}{48} + \frac{(x + 16) \cdot 3}{16 \cdot 3} = 1$

$y = - 4 - \frac{x}{4}$

Step 2.2.1.4.4

Multiply $16$ by $3$ .

$\frac{x \cdot 8}{48} + \frac{(x + 16) \cdot 3}{48} = 1$

$y = - 4 - \frac{x}{4}$

$\frac{x \cdot 8}{48} + \frac{(x + 16) \cdot 3}{48} = 1$

$y = - 4 - \frac{x}{4}$

Step 2.2.1.5

Combine the numerators over the common denominator.

$\frac{x \cdot 8 + (x + 16) \cdot 3}{48} = 1$

$y = - 4 - \frac{x}{4}$

Step 2.2.1.6

Simplify the numerator.

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Step 2.2.1.6.1

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

$\frac{8 \cdot x + (x + 16) \cdot 3}{48} = 1$

$y = - 4 - \frac{x}{4}$

Step 2.2.1.6.2

Apply the distributive property.

$\frac{8 x + x \cdot 3 + 16 \cdot 3}{48} = 1$

$y = - 4 - \frac{x}{4}$

Step 2.2.1.6.3

Move $3$ to the left of $x$ .

$\frac{8 x + 3 \cdot x + 16 \cdot 3}{48} = 1$

$y = - 4 - \frac{x}{4}$

Step 2.2.1.6.4

Multiply $16$ by $3$ .

$\frac{8 x + 3 \cdot x + 48}{48} = 1$

$y = - 4 - \frac{x}{4}$

Step 2.2.1.6.5

Add $8 x$ and $3 x$ .

$\frac{11 x + 48}{48} = 1$

$y = - 4 - \frac{x}{4}$

$\frac{11 x + 48}{48} = 1$

$y = - 4 - \frac{x}{4}$

$\frac{11 x + 48}{48} = 1$

$y = - 4 - \frac{x}{4}$

$\frac{11 x + 48}{48} = 1$

$y = - 4 - \frac{x}{4}$

$\frac{11 x + 48}{48} = 1$

$y = - 4 - \frac{x}{4}$

Step 3

Solve for $x$ in $\frac{11 x + 48}{48} = 1$ .

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Step 3.1

Multiply both sides by $48$ .

$\frac{11 x + 48}{48} \cdot 48 = 1 \cdot 48$

$y = - 4 - \frac{x}{4}$

Step 3.2

Simplify.

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Step 3.2.1

Simplify the left side.

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Step 3.2.1.1

Cancel the common factor of $48$ .

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Step 3.2.1.1.1

Cancel the common factor.

$\frac{11 x + 48}{48} \cdot 48 = 1 \cdot 48$

$y = - 4 - \frac{x}{4}$

Step 3.2.1.1.2

Rewrite the expression.

$11 x + 48 = 1 \cdot 48$

$y = - 4 - \frac{x}{4}$

$11 x + 48 = 1 \cdot 48$

$y = - 4 - \frac{x}{4}$

$11 x + 48 = 1 \cdot 48$

$y = - 4 - \frac{x}{4}$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Multiply $48$ by $1$ .

$11 x + 48 = 48$

$y = - 4 - \frac{x}{4}$

$11 x + 48 = 48$

$y = - 4 - \frac{x}{4}$

$11 x + 48 = 48$

$y = - 4 - \frac{x}{4}$

Step 3.3

Solve for $x$ .

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Step 3.3.1

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

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Step 3.3.1.1

Subtract $48$ from both sides of the equation.

$11 x = 48 - 48$

$y = - 4 - \frac{x}{4}$

Step 3.3.1.2

Subtract $48$ from $48$ .

$11 x = 0$

$y = - 4 - \frac{x}{4}$

$11 x = 0$

$y = - 4 - \frac{x}{4}$

Step 3.3.2

Divide each term in $11 x = 0$ by $11$ and simplify.

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Step 3.3.2.1

Divide each term in $11 x = 0$ by $11$ .

$\frac{11 x}{11} = \frac{0}{11}$

$y = - 4 - \frac{x}{4}$

Step 3.3.2.2

Simplify the left side.

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Step 3.3.2.2.1

Cancel the common factor of $11$ .

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Step 3.3.2.2.1.1

Cancel the common factor.

$\frac{11 x}{11} = \frac{0}{11}$

$y = - 4 - \frac{x}{4}$

Step 3.3.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{0}{11}$

$y = - 4 - \frac{x}{4}$

$x = \frac{0}{11}$

$y = - 4 - \frac{x}{4}$

$x = \frac{0}{11}$

$y = - 4 - \frac{x}{4}$

Step 3.3.2.3

Simplify the right side.

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Step 3.3.2.3.1

Divide $0$ by $11$ .

$x = 0$

$y = - 4 - \frac{x}{4}$

$x = 0$

$y = - 4 - \frac{x}{4}$

$x = 0$

$y = - 4 - \frac{x}{4}$

$x = 0$

$y = - 4 - \frac{x}{4}$

$x = 0$

$y = - 4 - \frac{x}{4}$

Step 4

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

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Step 4.1

Replace all occurrences of $x$ in $y = - 4 - \frac{x}{4}$ with $0$ .

$y = - 4 - \frac{0}{4}$

$x = 0$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $- 4 - \frac{0}{4}$ .

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Step 4.2.1.1

Simplify each term.

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Step 4.2.1.1.1

Divide $0$ by $4$ .

$y = - 4 - 0$

$x = 0$

Step 4.2.1.1.2

Multiply $- 1$ by $0$ .

$y = - 4 + 0$

$x = 0$

$y = - 4 + 0$

$x = 0$

Step 4.2.1.2

Add $- 4$ and $0$ .

$y = - 4$

$x = 0$

$y = - 4$

$x = 0$

$y = - 4$

$x = 0$

$y = - 4$

$x = 0$

Step 5

The solution to the system is the complete set of ordered pairs that are valid solutions.

$(0, - 4)$

Step 6

The result can be shown in multiple forms.

Point Form:

$(0, - 4)$

Equation Form:

$x = 0, y = - 4$

Step 7