Finite Math Examples

$- 4 x - 9 y = 1$ , $- x + 2 y = - 4$

Step 1

Solve for $x$ in $- 4 x - 9 y = 1$ .

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Step 1.1

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

$- 4 x = 1 + 9 y$

$- x + 2 y = - 4$

Step 1.2

Divide each term in $- 4 x = 1 + 9 y$ by $- 4$ and simplify.

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Step 1.2.1

Divide each term in $- 4 x = 1 + 9 y$ by $- 4$ .

$\frac{- 4 x}{- 4} = \frac{1}{- 4} + \frac{9 y}{- 4}$

$- x + 2 y = - 4$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $- 4$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{- 4 x}{- 4} = \frac{1}{- 4} + \frac{9 y}{- 4}$

$- x + 2 y = - 4$

Step 1.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{1}{- 4} + \frac{9 y}{- 4}$

$- x + 2 y = - 4$

$x = \frac{1}{- 4} + \frac{9 y}{- 4}$

$- x + 2 y = - 4$

$x = \frac{1}{- 4} + \frac{9 y}{- 4}$

$- x + 2 y = - 4$

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

Move the negative in front of the fraction.

$x = - \frac{1}{4} + \frac{9 y}{- 4}$

$- x + 2 y = - 4$

Step 1.2.3.1.2

Move the negative in front of the fraction.

$x = - \frac{1}{4} - \frac{9 y}{4}$

$- x + 2 y = - 4$

$x = - \frac{1}{4} - \frac{9 y}{4}$

$- x + 2 y = - 4$

$x = - \frac{1}{4} - \frac{9 y}{4}$

$- x + 2 y = - 4$

$x = - \frac{1}{4} - \frac{9 y}{4}$

$- x + 2 y = - 4$

$x = - \frac{1}{4} - \frac{9 y}{4}$

$- x + 2 y = - 4$

Step 2

Replace all occurrences of $x$ with $- \frac{1}{4} - \frac{9 y}{4}$ in each equation.

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Step 2.1

Replace all occurrences of $x$ in $- x + 2 y = - 4$ with $- \frac{1}{4} - \frac{9 y}{4}$ .

$- (- \frac{1}{4} - \frac{9 y}{4}) + 2 y = - 4$

$x = - \frac{1}{4} - \frac{9 y}{4}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $- (- \frac{1}{4} - \frac{9 y}{4}) + 2 y$ .

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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.

$\frac{1}{4} + \frac{9 y}{4} + 2 y = - 4$

$x = - \frac{1}{4} - \frac{9 y}{4}$

Step 2.2.1.1.2

Multiply $- - \frac{1}{4}$ .

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Step 2.2.1.1.2.1

Multiply $- 1$ by $- 1$ .

$1 (\frac{1}{4}) + \frac{9 y}{4} + 2 y = - 4$

$x = - \frac{1}{4} - \frac{9 y}{4}$

Step 2.2.1.1.2.2

Multiply $\frac{1}{4}$ by $1$ .

$\frac{1}{4} + \frac{9 y}{4} + 2 y = - 4$

$x = - \frac{1}{4} - \frac{9 y}{4}$

$\frac{1}{4} + \frac{9 y}{4} + 2 y = - 4$

$x = - \frac{1}{4} - \frac{9 y}{4}$

Step 2.2.1.1.3

Multiply $- - \frac{9 y}{4}$ .

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Step 2.2.1.1.3.1

Multiply $- 1$ by $- 1$ .

$\frac{1}{4} + 1 (\frac{9 y}{4}) + 2 y = - 4$

$x = - \frac{1}{4} - \frac{9 y}{4}$

Step 2.2.1.1.3.2

Multiply $\frac{9 y}{4}$ by $1$ .

$\frac{1}{4} + \frac{9 y}{4} + 2 y = - 4$

$x = - \frac{1}{4} - \frac{9 y}{4}$

$\frac{1}{4} + \frac{9 y}{4} + 2 y = - 4$

$x = - \frac{1}{4} - \frac{9 y}{4}$

$\frac{1}{4} + \frac{9 y}{4} + 2 y = - 4$

$x = - \frac{1}{4} - \frac{9 y}{4}$

Step 2.2.1.2

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

$\frac{1}{4} + \frac{9 y}{4} + 2 y \cdot \frac{4}{4} = - 4$

$x = - \frac{1}{4} - \frac{9 y}{4}$

Step 2.2.1.3

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

$\frac{1}{4} + \frac{9 y}{4} + \frac{2 y \cdot 4}{4} = - 4$

$x = - \frac{1}{4} - \frac{9 y}{4}$

Step 2.2.1.4

Combine the numerators over the common denominator.

$\frac{1}{4} + \frac{9 y + 2 y \cdot 4}{4} = - 4$

$x = - \frac{1}{4} - \frac{9 y}{4}$

Step 2.2.1.5

Combine the numerators over the common denominator.

$\frac{1 + 9 y + 2 y \cdot 4}{4} = - 4$

$x = - \frac{1}{4} - \frac{9 y}{4}$

Step 2.2.1.6

Multiply $4$ by $2$ .

$\frac{1 + 9 y + 8 y}{4} = - 4$

$x = - \frac{1}{4} - \frac{9 y}{4}$

Step 2.2.1.7

Add $9 y$ and $8 y$ .

$\frac{1 + 17 y}{4} = - 4$

$x = - \frac{1}{4} - \frac{9 y}{4}$

$\frac{1 + 17 y}{4} = - 4$

$x = - \frac{1}{4} - \frac{9 y}{4}$

$\frac{1 + 17 y}{4} = - 4$

$x = - \frac{1}{4} - \frac{9 y}{4}$

$\frac{1 + 17 y}{4} = - 4$

$x = - \frac{1}{4} - \frac{9 y}{4}$

Step 3

Solve for $y$ in $\frac{1 + 17 y}{4} = - 4$ .

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Step 3.1

Multiply both sides by $4$ .

$\frac{1 + 17 y}{4} \cdot 4 = - 4 \cdot 4$

$x = - \frac{1}{4} - \frac{9 y}{4}$

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

Simplify $\frac{1 + 17 y}{4} \cdot 4$ .

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Step 3.2.1.1.1

Cancel the common factor of $4$ .

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Step 3.2.1.1.1.1

Cancel the common factor.

$\frac{1 + 17 y}{4} \cdot 4 = - 4 \cdot 4$

$x = - \frac{1}{4} - \frac{9 y}{4}$

Step 3.2.1.1.1.2

Rewrite the expression.

$1 + 17 y = - 4 \cdot 4$

$x = - \frac{1}{4} - \frac{9 y}{4}$

$1 + 17 y = - 4 \cdot 4$

$x = - \frac{1}{4} - \frac{9 y}{4}$

Step 3.2.1.1.2

Reorder $1$ and $17 y$ .

$17 y + 1 = - 4 \cdot 4$

$x = - \frac{1}{4} - \frac{9 y}{4}$

$17 y + 1 = - 4 \cdot 4$

$x = - \frac{1}{4} - \frac{9 y}{4}$

$17 y + 1 = - 4 \cdot 4$

$x = - \frac{1}{4} - \frac{9 y}{4}$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Multiply $- 4$ by $4$ .

$17 y + 1 = - 16$

$x = - \frac{1}{4} - \frac{9 y}{4}$

$17 y + 1 = - 16$

$x = - \frac{1}{4} - \frac{9 y}{4}$

$17 y + 1 = - 16$

$x = - \frac{1}{4} - \frac{9 y}{4}$

Step 3.3

Solve for $y$ .

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Step 3.3.1

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

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Step 3.3.1.1

Subtract $1$ from both sides of the equation.

$17 y = - 16 - 1$

$x = - \frac{1}{4} - \frac{9 y}{4}$

Step 3.3.1.2

Subtract $1$ from $- 16$ .

$17 y = - 17$

$x = - \frac{1}{4} - \frac{9 y}{4}$

$17 y = - 17$

$x = - \frac{1}{4} - \frac{9 y}{4}$

Step 3.3.2

Divide each term in $17 y = - 17$ by $17$ and simplify.

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Step 3.3.2.1

Divide each term in $17 y = - 17$ by $17$ .

$\frac{17 y}{17} = \frac{- 17}{17}$

$x = - \frac{1}{4} - \frac{9 y}{4}$

Step 3.3.2.2

Simplify the left side.

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Step 3.3.2.2.1

Cancel the common factor of $17$ .

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Step 3.3.2.2.1.1

Cancel the common factor.

$\frac{17 y}{17} = \frac{- 17}{17}$

$x = - \frac{1}{4} - \frac{9 y}{4}$

Step 3.3.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 17}{17}$

$x = - \frac{1}{4} - \frac{9 y}{4}$

$y = \frac{- 17}{17}$

$x = - \frac{1}{4} - \frac{9 y}{4}$

$y = \frac{- 17}{17}$

$x = - \frac{1}{4} - \frac{9 y}{4}$

Step 3.3.2.3

Simplify the right side.

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Step 3.3.2.3.1

Divide $- 17$ by $17$ .

$y = - 1$

$x = - \frac{1}{4} - \frac{9 y}{4}$

$y = - 1$

$x = - \frac{1}{4} - \frac{9 y}{4}$

$y = - 1$

$x = - \frac{1}{4} - \frac{9 y}{4}$

$y = - 1$

$x = - \frac{1}{4} - \frac{9 y}{4}$

$y = - 1$

$x = - \frac{1}{4} - \frac{9 y}{4}$

Step 4

Replace all occurrences of $y$ with $- 1$ in each equation.

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Step 4.1

Replace all occurrences of $y$ in $x = - \frac{1}{4} - \frac{9 y}{4}$ with $- 1$ .

$x = - \frac{1}{4} - \frac{9 (- 1)}{4}$

$y = - 1$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $- \frac{1}{4} - \frac{9 (- 1)}{4}$ .

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Step 4.2.1.1

Combine the numerators over the common denominator.

$x = \frac{- 1 - 9 \cdot - 1}{4}$

$y = - 1$

Step 4.2.1.2

Simplify the expression.

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Step 4.2.1.2.1

Multiply $- 9$ by $- 1$ .

$x = \frac{- 1 + 9}{4}$

$y = - 1$

Step 4.2.1.2.2

Add $- 1$ and $9$ .

$x = \frac{8}{4}$

$y = - 1$

Step 4.2.1.2.3

Divide $8$ by $4$ .

$x = 2$

$y = - 1$

$x = 2$

$y = - 1$

$x = 2$

$y = - 1$

$x = 2$

$y = - 1$

$x = 2$

$y = - 1$

Step 5

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

$(2, - 1)$

Step 6

The result can be shown in multiple forms.

Point Form:

$(2, - 1)$

Equation Form:

$x = 2, y = - 1$

Step 7