Finite Math Examples

$9 x - 9 y = 36$ , $7 y - 3 x = - 14$

Step 1

Solve for $x$ in $9 x - 9 y = 36$ .

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Step 1.1

Add $9 y$ to both sides of the equation.

$9 x = 36 + 9 y$

$7 y - 3 x = - 14$

Step 1.2

Divide each term in $9 x = 36 + 9 y$ by $9$ and simplify.

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Step 1.2.1

Divide each term in $9 x = 36 + 9 y$ by $9$ .

$\frac{9 x}{9} = \frac{36}{9} + \frac{9 y}{9}$

$7 y - 3 x = - 14$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $9$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{9 x}{9} = \frac{36}{9} + \frac{9 y}{9}$

$7 y - 3 x = - 14$

Step 1.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{36}{9} + \frac{9 y}{9}$

$7 y - 3 x = - 14$

$x = \frac{36}{9} + \frac{9 y}{9}$

$7 y - 3 x = - 14$

$x = \frac{36}{9} + \frac{9 y}{9}$

$7 y - 3 x = - 14$

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

Simplify each term.

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Step 1.2.3.1.1

Divide $36$ by $9$ .

$x = 4 + \frac{9 y}{9}$

$7 y - 3 x = - 14$

Step 1.2.3.1.2

Cancel the common factor of $9$ .

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Step 1.2.3.1.2.1

Cancel the common factor.

$x = 4 + \frac{9 y}{9}$

$7 y - 3 x = - 14$

Step 1.2.3.1.2.2

Divide $y$ by $1$ .

$x = 4 + y$

$7 y - 3 x = - 14$

$x = 4 + y$

$7 y - 3 x = - 14$

$x = 4 + y$

$7 y - 3 x = - 14$

$x = 4 + y$

$7 y - 3 x = - 14$

$x = 4 + y$

$7 y - 3 x = - 14$

$x = 4 + y$

$7 y - 3 x = - 14$

Step 2

Replace all occurrences of $x$ with $4 + y$ in each equation.

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Step 2.1

Replace all occurrences of $x$ in $7 y - 3 x = - 14$ with $4 + y$ .

$7 y - 3 (4 + y) = - 14$

$x = 4 + y$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $7 y - 3 (4 + y)$ .

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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.

$7 y - 3 \cdot 4 - 3 y = - 14$

$x = 4 + y$

Step 2.2.1.1.2

Multiply $- 3$ by $4$ .

$7 y - 12 - 3 y = - 14$

$x = 4 + y$

$7 y - 12 - 3 y = - 14$

$x = 4 + y$

Step 2.2.1.2

Subtract $3 y$ from $7 y$ .

$4 y - 12 = - 14$

$x = 4 + y$

$4 y - 12 = - 14$

$x = 4 + y$

$4 y - 12 = - 14$

$x = 4 + y$

$4 y - 12 = - 14$

$x = 4 + y$

Step 3

Solve for $y$ in $4 y - 12 = - 14$ .

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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

Add $12$ to both sides of the equation.

$4 y = - 14 + 12$

$x = 4 + y$

Step 3.1.2

Add $- 14$ and $12$ .

$4 y = - 2$

$x = 4 + y$

$4 y = - 2$

$x = 4 + y$

Step 3.2

Divide each term in $4 y = - 2$ by $4$ and simplify.

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Step 3.2.1

Divide each term in $4 y = - 2$ by $4$ .

$\frac{4 y}{4} = \frac{- 2}{4}$

$x = 4 + y$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $4$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{4 y}{4} = \frac{- 2}{4}$

$x = 4 + y$

Step 3.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 2}{4}$

$x = 4 + y$

$y = \frac{- 2}{4}$

$x = 4 + y$

$y = \frac{- 2}{4}$

$x = 4 + y$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Cancel the common factor of $- 2$ and $4$ .

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Step 3.2.3.1.1

Factor $2$ out of $- 2$ .

$y = \frac{2 (- 1)}{4}$

$x = 4 + y$

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 $4$ .

$y = \frac{2 \cdot - 1}{2 \cdot 2}$

$x = 4 + y$

Step 3.2.3.1.2.2

Cancel the common factor.

$y = \frac{2 \cdot - 1}{2 \cdot 2}$

$x = 4 + y$

Step 3.2.3.1.2.3

Rewrite the expression.

$y = \frac{- 1}{2}$

$x = 4 + y$

$y = \frac{- 1}{2}$

$x = 4 + y$

$y = \frac{- 1}{2}$

$x = 4 + y$

Step 3.2.3.2

Move the negative in front of the fraction.

$y = - \frac{1}{2}$

$x = 4 + y$

$y = - \frac{1}{2}$

$x = 4 + y$

$y = - \frac{1}{2}$

$x = 4 + y$

$y = - \frac{1}{2}$

$x = 4 + y$

Step 4

Replace all occurrences of $y$ with $- \frac{1}{2}$ in each equation.

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Step 4.1

Replace all occurrences of $y$ in $x = 4 + y$ with $- \frac{1}{2}$ .

$x = 4 - \frac{1}{2}$

$y = - \frac{1}{2}$

Step 4.2

Simplify $x = 4 - \frac{1}{2}$ .

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Step 4.2.1

Simplify the left side.

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Step 4.2.1.1

Remove parentheses.

$x = 4 - \frac{1}{2}$

$y = - \frac{1}{2}$

$x = 4 - \frac{1}{2}$

$y = - \frac{1}{2}$

Step 4.2.2

Simplify the right side.

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Step 4.2.2.1

Simplify $4 - \frac{1}{2}$ .

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Step 4.2.2.1.1

To write $4$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$x = 4 \cdot \frac{2}{2} - \frac{1}{2}$

$y = - \frac{1}{2}$

Step 4.2.2.1.2

Combine $4$ and $\frac{2}{2}$ .

$x = \frac{4 \cdot 2}{2} - \frac{1}{2}$

$y = - \frac{1}{2}$

Step 4.2.2.1.3

Combine the numerators over the common denominator.

$x = \frac{4 \cdot 2 - 1}{2}$

$y = - \frac{1}{2}$

Step 4.2.2.1.4

Simplify the numerator.

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Step 4.2.2.1.4.1

Multiply $4$ by $2$ .

$x = \frac{8 - 1}{2}$

$y = - \frac{1}{2}$

Step 4.2.2.1.4.2

Subtract $1$ from $8$ .

$x = \frac{7}{2}$

$y = - \frac{1}{2}$

$x = \frac{7}{2}$

$y = - \frac{1}{2}$

$x = \frac{7}{2}$

$y = - \frac{1}{2}$

$x = \frac{7}{2}$

$y = - \frac{1}{2}$

$x = \frac{7}{2}$

$y = - \frac{1}{2}$

$x = \frac{7}{2}$

$y = - \frac{1}{2}$

Step 5

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

$(\frac{7}{2}, - \frac{1}{2})$

Step 6

The result can be shown in multiple forms.

Point Form:

$(\frac{7}{2}, - \frac{1}{2})$

Equation Form:

$x = \frac{7}{2}, y = - \frac{1}{2}$

Step 7