Finite Math Examples

$5 x + 2 y - 4 = 0$ , $3 x - y - 9 = 0$

Step 1

Solve for $x$ in $5 x + 2 y - 4 = 0$ .

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Step 1.1

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

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Step 1.1.1

Subtract $2 y$ from both sides of the equation.

$5 x - 4 = - 2 y$

$3 x - y - 9 = 0$

Step 1.1.2

Add $4$ to both sides of the equation.

$5 x = - 2 y + 4$

$3 x - y - 9 = 0$

$5 x = - 2 y + 4$

$3 x - y - 9 = 0$

Step 1.2

Divide each term in $5 x = - 2 y + 4$ by $5$ and simplify.

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Step 1.2.1

Divide each term in $5 x = - 2 y + 4$ by $5$ .

$\frac{5 x}{5} = \frac{- 2 y}{5} + \frac{4}{5}$

$3 x - y - 9 = 0$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $5$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{5 x}{5} = \frac{- 2 y}{5} + \frac{4}{5}$

$3 x - y - 9 = 0$

Step 1.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 2 y}{5} + \frac{4}{5}$

$3 x - y - 9 = 0$

$x = \frac{- 2 y}{5} + \frac{4}{5}$

$3 x - y - 9 = 0$

$x = \frac{- 2 y}{5} + \frac{4}{5}$

$3 x - y - 9 = 0$

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

Move the negative in front of the fraction.

$x = - \frac{2 y}{5} + \frac{4}{5}$

$3 x - y - 9 = 0$

$x = - \frac{2 y}{5} + \frac{4}{5}$

$3 x - y - 9 = 0$

$x = - \frac{2 y}{5} + \frac{4}{5}$

$3 x - y - 9 = 0$

$x = - \frac{2 y}{5} + \frac{4}{5}$

$3 x - y - 9 = 0$

Step 2

Replace all occurrences of $x$ with $- \frac{2 y}{5} + \frac{4}{5}$ in each equation.

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Step 2.1

Replace all occurrences of $x$ in $3 x - y - 9 = 0$ with $- \frac{2 y}{5} + \frac{4}{5}$ .

$3 (- \frac{2 y}{5} + \frac{4}{5}) - y - 9 = 0$

$x = - \frac{2 y}{5} + \frac{4}{5}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $3 (- \frac{2 y}{5} + \frac{4}{5}) - y - 9$ .

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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.

$3 (- \frac{2 y}{5}) + 3 (\frac{4}{5}) - y - 9 = 0$

$x = - \frac{2 y}{5} + \frac{4}{5}$

Step 2.2.1.1.2

Multiply $3 (- \frac{2 y}{5})$ .

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Step 2.2.1.1.2.1

Multiply $- 1$ by $3$ .

$- 3 \frac{2 y}{5} + 3 (\frac{4}{5}) - y - 9 = 0$

$x = - \frac{2 y}{5} + \frac{4}{5}$

Step 2.2.1.1.2.2

Combine $- 3$ and $\frac{2 y}{5}$ .

$\frac{- 3 (2 y)}{5} + 3 (\frac{4}{5}) - y - 9 = 0$

$x = - \frac{2 y}{5} + \frac{4}{5}$

Step 2.2.1.1.2.3

Multiply $2$ by $- 3$ .

$\frac{- 6 y}{5} + 3 (\frac{4}{5}) - y - 9 = 0$

$x = - \frac{2 y}{5} + \frac{4}{5}$

$\frac{- 6 y}{5} + 3 (\frac{4}{5}) - y - 9 = 0$

$x = - \frac{2 y}{5} + \frac{4}{5}$

Step 2.2.1.1.3

Multiply $3 (\frac{4}{5})$ .

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Step 2.2.1.1.3.1

Combine $3$ and $\frac{4}{5}$ .

$\frac{- 6 y}{5} + \frac{3 \cdot 4}{5} - y - 9 = 0$

$x = - \frac{2 y}{5} + \frac{4}{5}$

Step 2.2.1.1.3.2

Multiply $3$ by $4$ .

$\frac{- 6 y}{5} + \frac{12}{5} - y - 9 = 0$

$x = - \frac{2 y}{5} + \frac{4}{5}$

$\frac{- 6 y}{5} + \frac{12}{5} - y - 9 = 0$

$x = - \frac{2 y}{5} + \frac{4}{5}$

Step 2.2.1.1.4

Move the negative in front of the fraction.

$- \frac{6 y}{5} + \frac{12}{5} - y - 9 = 0$

$x = - \frac{2 y}{5} + \frac{4}{5}$

$- \frac{6 y}{5} + \frac{12}{5} - y - 9 = 0$

$x = - \frac{2 y}{5} + \frac{4}{5}$

Step 2.2.1.2

To write $- y$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$- \frac{6 y}{5} - y \cdot \frac{5}{5} + \frac{12}{5} - 9 = 0$

$x = - \frac{2 y}{5} + \frac{4}{5}$

Step 2.2.1.3

Combine $- y$ and $\frac{5}{5}$ .

$- \frac{6 y}{5} + \frac{- y \cdot 5}{5} + \frac{12}{5} - 9 = 0$

$x = - \frac{2 y}{5} + \frac{4}{5}$

Step 2.2.1.4

Combine the numerators over the common denominator.

$\frac{- 6 y - y \cdot 5}{5} + \frac{12}{5} - 9 = 0$

$x = - \frac{2 y}{5} + \frac{4}{5}$

Step 2.2.1.5

Combine the numerators over the common denominator.

$- 9 + \frac{- 6 y - y \cdot 5 + 12}{5} = 0$

$x = - \frac{2 y}{5} + \frac{4}{5}$

Step 2.2.1.6

Multiply $5$ by $- 1$ .

$- 9 + \frac{- 6 y - 5 y + 12}{5} = 0$

$x = - \frac{2 y}{5} + \frac{4}{5}$

Step 2.2.1.7

Subtract $5 y$ from $- 6 y$ .

$- 9 + \frac{- 11 y + 12}{5} = 0$

$x = - \frac{2 y}{5} + \frac{4}{5}$

Step 2.2.1.8

To write $- 9$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$- 9 \cdot \frac{5}{5} + \frac{- 11 y + 12}{5} = 0$

$x = - \frac{2 y}{5} + \frac{4}{5}$

Step 2.2.1.9

Simplify terms.

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Step 2.2.1.9.1

Combine $- 9$ and $\frac{5}{5}$ .

$\frac{- 9 \cdot 5}{5} + \frac{- 11 y + 12}{5} = 0$

$x = - \frac{2 y}{5} + \frac{4}{5}$

Step 2.2.1.9.2

Combine the numerators over the common denominator.

$\frac{- 9 \cdot 5 - 11 y + 12}{5} = 0$

$x = - \frac{2 y}{5} + \frac{4}{5}$

$\frac{- 9 \cdot 5 - 11 y + 12}{5} = 0$

$x = - \frac{2 y}{5} + \frac{4}{5}$

Step 2.2.1.10

Simplify the numerator.

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Step 2.2.1.10.1

Multiply $- 9$ by $5$ .

$\frac{- 45 - 11 y + 12}{5} = 0$

$x = - \frac{2 y}{5} + \frac{4}{5}$

Step 2.2.1.10.2

Add $- 45$ and $12$ .

$\frac{- 11 y - 33}{5} = 0$

$x = - \frac{2 y}{5} + \frac{4}{5}$

Step 2.2.1.10.3

Factor $11$ out of $- 11 y - 33$ .

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Step 2.2.1.10.3.1

Factor $11$ out of $- 11 y$ .

$\frac{11 (- y) - 33}{5} = 0$

$x = - \frac{2 y}{5} + \frac{4}{5}$

Step 2.2.1.10.3.2

Factor $11$ out of $- 33$ .

$\frac{11 (- y) + 11 (- 3)}{5} = 0$

$x = - \frac{2 y}{5} + \frac{4}{5}$

Step 2.2.1.10.3.3

Factor $11$ out of $11 (- y) + 11 (- 3)$ .

$\frac{11 (- y - 3)}{5} = 0$

$x = - \frac{2 y}{5} + \frac{4}{5}$

$\frac{11 (- y - 3)}{5} = 0$

$x = - \frac{2 y}{5} + \frac{4}{5}$

$\frac{11 (- y - 3)}{5} = 0$

$x = - \frac{2 y}{5} + \frac{4}{5}$

Step 2.2.1.11

Simplify with factoring out.

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Step 2.2.1.11.1

Factor $- 1$ out of $- y$ .

$\frac{11 (- (y) - 3)}{5} = 0$

$x = - \frac{2 y}{5} + \frac{4}{5}$

Step 2.2.1.11.2

Rewrite $- 3$ as $- 1 (3)$ .

$\frac{11 (- (y) - 1 \cdot 3)}{5} = 0$

$x = - \frac{2 y}{5} + \frac{4}{5}$

Step 2.2.1.11.3

Factor $- 1$ out of $- (y) - 1 (3)$ .

$\frac{11 (- (y + 3))}{5} = 0$

$x = - \frac{2 y}{5} + \frac{4}{5}$

Step 2.2.1.11.4

Simplify the expression.

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Step 2.2.1.11.4.1

Rewrite $- (y + 3)$ as $- 1 (y + 3)$ .

$\frac{11 (- 1 (y + 3))}{5} = 0$

$x = - \frac{2 y}{5} + \frac{4}{5}$

Step 2.2.1.11.4.2

Move the negative in front of the fraction.

$- \frac{11 (y + 3)}{5} = 0$

$x = - \frac{2 y}{5} + \frac{4}{5}$

$- \frac{11 (y + 3)}{5} = 0$

$x = - \frac{2 y}{5} + \frac{4}{5}$

$- \frac{11 (y + 3)}{5} = 0$

$x = - \frac{2 y}{5} + \frac{4}{5}$

$- \frac{11 (y + 3)}{5} = 0$

$x = - \frac{2 y}{5} + \frac{4}{5}$

$- \frac{11 (y + 3)}{5} = 0$

$x = - \frac{2 y}{5} + \frac{4}{5}$

$- \frac{11 (y + 3)}{5} = 0$

$x = - \frac{2 y}{5} + \frac{4}{5}$

Step 3

Solve for $y$ in $- \frac{11 (y + 3)}{5} = 0$ .

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Step 3.1

Set the numerator equal to zero.

$11 (y + 3) = 0$

$x = - \frac{2 y}{5} + \frac{4}{5}$

Step 3.2

Solve the equation for $y$ .

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Step 3.2.1

Divide each term in $11 (y + 3) = 0$ by $11$ and simplify.

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Step 3.2.1.1

Divide each term in $11 (y + 3) = 0$ by $11$ .

$\frac{11 (y + 3)}{11} = \frac{0}{11}$

$x = - \frac{2 y}{5} + \frac{4}{5}$

Step 3.2.1.2

Simplify the left side.

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Step 3.2.1.2.1

Cancel the common factor of $11$ .

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Step 3.2.1.2.1.1

Cancel the common factor.

$\frac{11 (y + 3)}{11} = \frac{0}{11}$

$x = - \frac{2 y}{5} + \frac{4}{5}$

Step 3.2.1.2.1.2

Divide $y + 3$ by $1$ .

$y + 3 = \frac{0}{11}$

$x = - \frac{2 y}{5} + \frac{4}{5}$

$y + 3 = \frac{0}{11}$

$x = - \frac{2 y}{5} + \frac{4}{5}$

$y + 3 = \frac{0}{11}$

$x = - \frac{2 y}{5} + \frac{4}{5}$

Step 3.2.1.3

Simplify the right side.

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Step 3.2.1.3.1

Divide $0$ by $11$ .

$y + 3 = 0$

$x = - \frac{2 y}{5} + \frac{4}{5}$

$y + 3 = 0$

$x = - \frac{2 y}{5} + \frac{4}{5}$

$y + 3 = 0$

$x = - \frac{2 y}{5} + \frac{4}{5}$

Step 3.2.2

Subtract $3$ from both sides of the equation.

$y = - 3$

$x = - \frac{2 y}{5} + \frac{4}{5}$

$y = - 3$

$x = - \frac{2 y}{5} + \frac{4}{5}$

$y = - 3$

$x = - \frac{2 y}{5} + \frac{4}{5}$

Step 4

Replace all occurrences of $y$ with $- 3$ in each equation.

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Step 4.1

Replace all occurrences of $y$ in $x = - \frac{2 y}{5} + \frac{4}{5}$ with $- 3$ .

$x = - \frac{2 (- 3)}{5} + \frac{4}{5}$

$y = - 3$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $- \frac{2 (- 3)}{5} + \frac{4}{5}$ .

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Step 4.2.1.1

Combine the numerators over the common denominator.

$x = \frac{- 2 \cdot - 3 + 4}{5}$

$y = - 3$

Step 4.2.1.2

Simplify the expression.

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Step 4.2.1.2.1

Multiply $- 2$ by $- 3$ .

$x = \frac{6 + 4}{5}$

$y = - 3$

Step 4.2.1.2.2

Add $6$ and $4$ .

$x = \frac{10}{5}$

$y = - 3$

Step 4.2.1.2.3

Divide $10$ by $5$ .

$x = 2$

$y = - 3$

$x = 2$

$y = - 3$

$x = 2$

$y = - 3$

$x = 2$

$y = - 3$

$x = 2$

$y = - 3$

Step 5

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

$(2, - 3)$

Step 6

The result can be shown in multiple forms.

Point Form:

$(2, - 3)$

Equation Form:

$x = 2, y = - 3$

Step 7