Finite Math Examples

$5 (4 x + y) - (x - 3 y) = - 5$ , $2 (x - 2 y) + (2 x - y) = 19$

Step 1

Solve for $x$ in $5 (4 x + y) - x + 3 y = - 5$ .

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Step 1.1

Simplify $5 (4 x + y) - x + 3 y$ .

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Step 1.1.1

Simplify each term.

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Step 1.1.1.1

Apply the distributive property.

$5 (4 x) + 5 y - x + 3 y = - 5$

$2 (x - 2 y) + 2 x - y = 19$

Step 1.1.1.2

Multiply $4$ by $5$ .

$20 x + 5 y - x + 3 y = - 5$

$2 (x - 2 y) + 2 x - y = 19$

$20 x + 5 y - x + 3 y = - 5$

$2 (x - 2 y) + 2 x - y = 19$

Step 1.1.2

Simplify by adding terms.

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Step 1.1.2.1

Subtract $x$ from $20 x$ .

$19 x + 5 y + 3 y = - 5$

$2 (x - 2 y) + 2 x - y = 19$

Step 1.1.2.2

Add $5 y$ and $3 y$ .

$19 x + 8 y = - 5$

$2 (x - 2 y) + 2 x - y = 19$

$19 x + 8 y = - 5$

$2 (x - 2 y) + 2 x - y = 19$

$19 x + 8 y = - 5$

$2 (x - 2 y) + 2 x - y = 19$

Step 1.2

Subtract $8 y$ from both sides of the equation.

$19 x = - 5 - 8 y$

$2 (x - 2 y) + 2 x - y = 19$

Step 1.3

Divide each term in $19 x = - 5 - 8 y$ by $19$ and simplify.

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Step 1.3.1

Divide each term in $19 x = - 5 - 8 y$ by $19$ .

$\frac{19 x}{19} = \frac{- 5}{19} + \frac{- 8 y}{19}$

$2 (x - 2 y) + 2 x - y = 19$

Step 1.3.2

Simplify the left side.

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Step 1.3.2.1

Cancel the common factor of $19$ .

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Step 1.3.2.1.1

Cancel the common factor.

$\frac{19 x}{19} = \frac{- 5}{19} + \frac{- 8 y}{19}$

$2 (x - 2 y) + 2 x - y = 19$

Step 1.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 5}{19} + \frac{- 8 y}{19}$

$2 (x - 2 y) + 2 x - y = 19$

$x = \frac{- 5}{19} + \frac{- 8 y}{19}$

$2 (x - 2 y) + 2 x - y = 19$

$x = \frac{- 5}{19} + \frac{- 8 y}{19}$

$2 (x - 2 y) + 2 x - y = 19$

Step 1.3.3

Simplify the right side.

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Step 1.3.3.1

Simplify each term.

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Step 1.3.3.1.1

Move the negative in front of the fraction.

$x = - \frac{5}{19} + \frac{- 8 y}{19}$

$2 (x - 2 y) + 2 x - y = 19$

Step 1.3.3.1.2

Move the negative in front of the fraction.

$x = - \frac{5}{19} - \frac{8 y}{19}$

$2 (x - 2 y) + 2 x - y = 19$

$x = - \frac{5}{19} - \frac{8 y}{19}$

$2 (x - 2 y) + 2 x - y = 19$

$x = - \frac{5}{19} - \frac{8 y}{19}$

$2 (x - 2 y) + 2 x - y = 19$

$x = - \frac{5}{19} - \frac{8 y}{19}$

$2 (x - 2 y) + 2 x - y = 19$

$x = - \frac{5}{19} - \frac{8 y}{19}$

$2 (x - 2 y) + 2 x - y = 19$

Step 2

Replace all occurrences of $x$ with $- \frac{5}{19} - \frac{8 y}{19}$ in each equation.

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Step 2.1

Replace all occurrences of $x$ in $2 (x - 2 y) + 2 x - y = 19$ with $- \frac{5}{19} - \frac{8 y}{19}$ .

$2 ((- \frac{5}{19} - \frac{8 y}{19}) - 2 y) + 2 (- \frac{5}{19} - \frac{8 y}{19}) - y = 19$

$x = - \frac{5}{19} - \frac{8 y}{19}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $2 ((- \frac{5}{19} - \frac{8 y}{19}) - 2 y) + 2 (- \frac{5}{19} - \frac{8 y}{19}) - y$ .

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Step 2.2.1.1

Simplify each term.

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Step 2.2.1.1.1

To write $- 2 y$ as a fraction with a common denominator, multiply by $\frac{19}{19}$ .

$2 (- \frac{5}{19} - \frac{8 y}{19} - 2 y \cdot \frac{19}{19}) + 2 (- \frac{5}{19} - \frac{8 y}{19}) - y = 19$

$x = - \frac{5}{19} - \frac{8 y}{19}$

Step 2.2.1.1.2

Combine $- 2 y$ and $\frac{19}{19}$ .

$2 (- \frac{5}{19} - \frac{8 y}{19} + \frac{- 2 y \cdot 19}{19}) + 2 (- \frac{5}{19} - \frac{8 y}{19}) - y = 19$

$x = - \frac{5}{19} - \frac{8 y}{19}$

Step 2.2.1.1.3

Combine the numerators over the common denominator.

$2 (- \frac{5}{19} + \frac{- 8 y - 2 y \cdot 19}{19}) + 2 (- \frac{5}{19} - \frac{8 y}{19}) - y = 19$

$x = - \frac{5}{19} - \frac{8 y}{19}$

Step 2.2.1.1.4

Combine the numerators over the common denominator.

$2 (\frac{- 5 - 8 y - 2 y \cdot 19}{19}) + 2 (- \frac{5}{19} - \frac{8 y}{19}) - y = 19$

$x = - \frac{5}{19} - \frac{8 y}{19}$

Step 2.2.1.1.5

Multiply $19$ by $- 2$ .

$2 (\frac{- 5 - 8 y - 38 y}{19}) + 2 (- \frac{5}{19} - \frac{8 y}{19}) - y = 19$

$x = - \frac{5}{19} - \frac{8 y}{19}$

Step 2.2.1.1.6

Subtract $38 y$ from $- 8 y$ .

$2 (\frac{- 5 - 46 y}{19}) + 2 (- \frac{5}{19} - \frac{8 y}{19}) - y = 19$

$x = - \frac{5}{19} - \frac{8 y}{19}$

Step 2.2.1.1.7

Combine $2$ and $\frac{- 5 - 46 y}{19}$ .

$\frac{2 (- 5 - 46 y)}{19} + 2 (- \frac{5}{19} - \frac{8 y}{19}) - y = 19$

$x = - \frac{5}{19} - \frac{8 y}{19}$

Step 2.2.1.1.8

Apply the distributive property.

$\frac{2 (- 5 - 46 y)}{19} + 2 (- \frac{5}{19}) + 2 (- \frac{8 y}{19}) - y = 19$

$x = - \frac{5}{19} - \frac{8 y}{19}$

Step 2.2.1.1.9

Multiply $2 (- \frac{5}{19})$ .

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Step 2.2.1.1.9.1

Multiply $- 1$ by $2$ .

$\frac{2 (- 5 - 46 y)}{19} - 2 (\frac{5}{19}) + 2 (- \frac{8 y}{19}) - y = 19$

$x = - \frac{5}{19} - \frac{8 y}{19}$

Step 2.2.1.1.9.2

Combine $- 2$ and $\frac{5}{19}$ .

$\frac{2 (- 5 - 46 y)}{19} + \frac{- 2 \cdot 5}{19} + 2 (- \frac{8 y}{19}) - y = 19$

$x = - \frac{5}{19} - \frac{8 y}{19}$

Step 2.2.1.1.9.3

Multiply $- 2$ by $5$ .

$\frac{2 (- 5 - 46 y)}{19} + \frac{- 10}{19} + 2 (- \frac{8 y}{19}) - y = 19$

$x = - \frac{5}{19} - \frac{8 y}{19}$

$\frac{2 (- 5 - 46 y)}{19} + \frac{- 10}{19} + 2 (- \frac{8 y}{19}) - y = 19$

$x = - \frac{5}{19} - \frac{8 y}{19}$

Step 2.2.1.1.10

Multiply $2 (- \frac{8 y}{19})$ .

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Step 2.2.1.1.10.1

Multiply $- 1$ by $2$ .

$\frac{2 (- 5 - 46 y)}{19} + \frac{- 10}{19} - 2 \frac{8 y}{19} - y = 19$

$x = - \frac{5}{19} - \frac{8 y}{19}$

Step 2.2.1.1.10.2

Combine $- 2$ and $\frac{8 y}{19}$ .

$\frac{2 (- 5 - 46 y)}{19} + \frac{- 10}{19} + \frac{- 2 (8 y)}{19} - y = 19$

$x = - \frac{5}{19} - \frac{8 y}{19}$

Step 2.2.1.1.10.3

Multiply $8$ by $- 2$ .

$\frac{2 (- 5 - 46 y)}{19} + \frac{- 10}{19} + \frac{- 16 y}{19} - y = 19$

$x = - \frac{5}{19} - \frac{8 y}{19}$

$\frac{2 (- 5 - 46 y)}{19} + \frac{- 10}{19} + \frac{- 16 y}{19} - y = 19$

$x = - \frac{5}{19} - \frac{8 y}{19}$

Step 2.2.1.1.11

Simplify each term.

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Step 2.2.1.1.11.1

Move the negative in front of the fraction.

$\frac{2 (- 5 - 46 y)}{19} - \frac{10}{19} + \frac{- 16 y}{19} - y = 19$

$x = - \frac{5}{19} - \frac{8 y}{19}$

Step 2.2.1.1.11.2

Move the negative in front of the fraction.

$\frac{2 (- 5 - 46 y)}{19} - \frac{10}{19} - \frac{16 y}{19} - y = 19$

$x = - \frac{5}{19} - \frac{8 y}{19}$

$\frac{2 (- 5 - 46 y)}{19} - \frac{10}{19} - \frac{16 y}{19} - y = 19$

$x = - \frac{5}{19} - \frac{8 y}{19}$

$\frac{2 (- 5 - 46 y)}{19} - \frac{10}{19} - \frac{16 y}{19} - y = 19$

$x = - \frac{5}{19} - \frac{8 y}{19}$

Step 2.2.1.2

Combine the numerators over the common denominator.

$- y + \frac{2 (- 5 - 46 y) - 10 - 16 y}{19} = 19$

$x = - \frac{5}{19} - \frac{8 y}{19}$

Step 2.2.1.3

Simplify each term.

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Step 2.2.1.3.1

Apply the distributive property.

$- y + \frac{2 \cdot - 5 + 2 (- 46 y) - 10 - 16 y}{19} = 19$

$x = - \frac{5}{19} - \frac{8 y}{19}$

Step 2.2.1.3.2

Multiply $2$ by $- 5$ .

$- y + \frac{- 10 + 2 (- 46 y) - 10 - 16 y}{19} = 19$

$x = - \frac{5}{19} - \frac{8 y}{19}$

Step 2.2.1.3.3

Multiply $- 46$ by $2$ .

$- y + \frac{- 10 - 92 y - 10 - 16 y}{19} = 19$

$x = - \frac{5}{19} - \frac{8 y}{19}$

$- y + \frac{- 10 - 92 y - 10 - 16 y}{19} = 19$

$x = - \frac{5}{19} - \frac{8 y}{19}$

Step 2.2.1.4

Simplify terms.

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Step 2.2.1.4.1

Subtract $10$ from $- 10$ .

$- y + \frac{- 92 y - 20 - 16 y}{19} = 19$

$x = - \frac{5}{19} - \frac{8 y}{19}$

Step 2.2.1.4.2

Subtract $16 y$ from $- 92 y$ .

$- y + \frac{- 108 y - 20}{19} = 19$

$x = - \frac{5}{19} - \frac{8 y}{19}$

Step 2.2.1.4.3

Factor $4$ out of $- 108 y - 20$ .

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Step 2.2.1.4.3.1

Factor $4$ out of $- 108 y$ .

$- y + \frac{4 (- 27 y) - 20}{19} = 19$

$x = - \frac{5}{19} - \frac{8 y}{19}$

Step 2.2.1.4.3.2

Factor $4$ out of $- 20$ .

$- y + \frac{4 (- 27 y) + 4 (- 5)}{19} = 19$

$x = - \frac{5}{19} - \frac{8 y}{19}$

Step 2.2.1.4.3.3

Factor $4$ out of $4 (- 27 y) + 4 (- 5)$ .

$- y + \frac{4 (- 27 y - 5)}{19} = 19$

$x = - \frac{5}{19} - \frac{8 y}{19}$

$- y + \frac{4 (- 27 y - 5)}{19} = 19$

$x = - \frac{5}{19} - \frac{8 y}{19}$

$- y + \frac{4 (- 27 y - 5)}{19} = 19$

$x = - \frac{5}{19} - \frac{8 y}{19}$

Step 2.2.1.5

To write $- y$ as a fraction with a common denominator, multiply by $\frac{19}{19}$ .

$- y \cdot \frac{19}{19} + \frac{4 (- 27 y - 5)}{19} = 19$

$x = - \frac{5}{19} - \frac{8 y}{19}$

Step 2.2.1.6

Simplify terms.

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Step 2.2.1.6.1

Combine $- y$ and $\frac{19}{19}$ .

$\frac{- y \cdot 19}{19} + \frac{4 (- 27 y - 5)}{19} = 19$

$x = - \frac{5}{19} - \frac{8 y}{19}$

Step 2.2.1.6.2

Combine the numerators over the common denominator.

$\frac{- y \cdot 19 + 4 (- 27 y - 5)}{19} = 19$

$x = - \frac{5}{19} - \frac{8 y}{19}$

$\frac{- y \cdot 19 + 4 (- 27 y - 5)}{19} = 19$

$x = - \frac{5}{19} - \frac{8 y}{19}$

Step 2.2.1.7

Simplify the numerator.

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Step 2.2.1.7.1

Multiply $19$ by $- 1$ .

$\frac{- 19 y + 4 (- 27 y - 5)}{19} = 19$

$x = - \frac{5}{19} - \frac{8 y}{19}$

Step 2.2.1.7.2

Apply the distributive property.

$\frac{- 19 y + 4 (- 27 y) + 4 \cdot - 5}{19} = 19$

$x = - \frac{5}{19} - \frac{8 y}{19}$

Step 2.2.1.7.3

Multiply $- 27$ by $4$ .

$\frac{- 19 y - 108 y + 4 \cdot - 5}{19} = 19$

$x = - \frac{5}{19} - \frac{8 y}{19}$

Step 2.2.1.7.4

Multiply $4$ by $- 5$ .

$\frac{- 19 y - 108 y - 20}{19} = 19$

$x = - \frac{5}{19} - \frac{8 y}{19}$

Step 2.2.1.7.5

Subtract $108 y$ from $- 19 y$ .

$\frac{- 127 y - 20}{19} = 19$

$x = - \frac{5}{19} - \frac{8 y}{19}$

$\frac{- 127 y - 20}{19} = 19$

$x = - \frac{5}{19} - \frac{8 y}{19}$

Step 2.2.1.8

Simplify with factoring out.

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Step 2.2.1.8.1

Factor $- 1$ out of $- 127 y$ .

$\frac{- (127 y) - 20}{19} = 19$

$x = - \frac{5}{19} - \frac{8 y}{19}$

Step 2.2.1.8.2

Rewrite $- 20$ as $- 1 (20)$ .

$\frac{- (127 y) - 1 \cdot 20}{19} = 19$

$x = - \frac{5}{19} - \frac{8 y}{19}$

Step 2.2.1.8.3

Factor $- 1$ out of $- (127 y) - 1 (20)$ .

$\frac{- (127 y + 20)}{19} = 19$

$x = - \frac{5}{19} - \frac{8 y}{19}$

Step 2.2.1.8.4

Simplify the expression.

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Step 2.2.1.8.4.1

Rewrite $- (127 y + 20)$ as $- 1 (127 y + 20)$ .

$\frac{- 1 (127 y + 20)}{19} = 19$

$x = - \frac{5}{19} - \frac{8 y}{19}$

Step 2.2.1.8.4.2

Move the negative in front of the fraction.

$- \frac{127 y + 20}{19} = 19$

$x = - \frac{5}{19} - \frac{8 y}{19}$

$- \frac{127 y + 20}{19} = 19$

$x = - \frac{5}{19} - \frac{8 y}{19}$

$- \frac{127 y + 20}{19} = 19$

$x = - \frac{5}{19} - \frac{8 y}{19}$

$- \frac{127 y + 20}{19} = 19$

$x = - \frac{5}{19} - \frac{8 y}{19}$

$- \frac{127 y + 20}{19} = 19$

$x = - \frac{5}{19} - \frac{8 y}{19}$

$- \frac{127 y + 20}{19} = 19$

$x = - \frac{5}{19} - \frac{8 y}{19}$

Step 3

Solve for $y$ in $- \frac{127 y + 20}{19} = 19$ .

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Step 3.1

Multiply both sides of the equation by $- 19$ .

$- 19 (- \frac{127 y + 20}{19}) = - 19 \cdot 19$

$x = - \frac{5}{19} - \frac{8 y}{19}$

Step 3.2

Simplify both sides of the equation.

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Step 3.2.1

Simplify the left side.

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Step 3.2.1.1

Simplify $- 19 (- \frac{127 y + 20}{19})$ .

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Step 3.2.1.1.1

Cancel the common factor of $19$ .

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Step 3.2.1.1.1.1

Move the leading negative in $- \frac{127 y + 20}{19}$ into the numerator.

$- 19 \frac{- (127 y + 20)}{19} = - 19 \cdot 19$

$x = - \frac{5}{19} - \frac{8 y}{19}$

Step 3.2.1.1.1.2

Factor $19$ out of $- 19$ .

$19 (- 1) (\frac{- (127 y + 20)}{19}) = - 19 \cdot 19$

$x = - \frac{5}{19} - \frac{8 y}{19}$

Step 3.2.1.1.1.3

Cancel the common factor.

$19 \cdot (- 1 \frac{- (127 y + 20)}{19}) = - 19 \cdot 19$

$x = - \frac{5}{19} - \frac{8 y}{19}$

Step 3.2.1.1.1.4

Rewrite the expression.

$127 y + 20 = - 19 \cdot 19$

$x = - \frac{5}{19} - \frac{8 y}{19}$

$127 y + 20 = - 19 \cdot 19$

$x = - \frac{5}{19} - \frac{8 y}{19}$

Step 3.2.1.1.2

Multiply.

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Step 3.2.1.1.2.1

Multiply $- 1$ by $- 1$ .

$1 (127 y + 20) = - 19 \cdot 19$

$x = - \frac{5}{19} - \frac{8 y}{19}$

Step 3.2.1.1.2.2

Multiply $127 y + 20$ by $1$ .

$127 y + 20 = - 19 \cdot 19$

$x = - \frac{5}{19} - \frac{8 y}{19}$

$127 y + 20 = - 19 \cdot 19$

$x = - \frac{5}{19} - \frac{8 y}{19}$

$127 y + 20 = - 19 \cdot 19$

$x = - \frac{5}{19} - \frac{8 y}{19}$

$127 y + 20 = - 19 \cdot 19$

$x = - \frac{5}{19} - \frac{8 y}{19}$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Multiply $- 19$ by $19$ .

$127 y + 20 = - 361$

$x = - \frac{5}{19} - \frac{8 y}{19}$

$127 y + 20 = - 361$

$x = - \frac{5}{19} - \frac{8 y}{19}$

$127 y + 20 = - 361$

$x = - \frac{5}{19} - \frac{8 y}{19}$

Step 3.3

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

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Step 3.3.1

Subtract $20$ from both sides of the equation.

$127 y = - 361 - 20$

$x = - \frac{5}{19} - \frac{8 y}{19}$

Step 3.3.2

Subtract $20$ from $- 361$ .

$127 y = - 381$

$x = - \frac{5}{19} - \frac{8 y}{19}$

$127 y = - 381$

$x = - \frac{5}{19} - \frac{8 y}{19}$

Step 3.4

Divide each term in $127 y = - 381$ by $127$ and simplify.

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Step 3.4.1

Divide each term in $127 y = - 381$ by $127$ .

$\frac{127 y}{127} = \frac{- 381}{127}$

$x = - \frac{5}{19} - \frac{8 y}{19}$

Step 3.4.2

Simplify the left side.

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Step 3.4.2.1

Cancel the common factor of $127$ .

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Step 3.4.2.1.1

Cancel the common factor.

$\frac{127 y}{127} = \frac{- 381}{127}$

$x = - \frac{5}{19} - \frac{8 y}{19}$

Step 3.4.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 381}{127}$

$x = - \frac{5}{19} - \frac{8 y}{19}$

$y = \frac{- 381}{127}$

$x = - \frac{5}{19} - \frac{8 y}{19}$

$y = \frac{- 381}{127}$

$x = - \frac{5}{19} - \frac{8 y}{19}$

Step 3.4.3

Simplify the right side.

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Step 3.4.3.1

Divide $- 381$ by $127$ .

$y = - 3$

$x = - \frac{5}{19} - \frac{8 y}{19}$

$y = - 3$

$x = - \frac{5}{19} - \frac{8 y}{19}$

$y = - 3$

$x = - \frac{5}{19} - \frac{8 y}{19}$

$y = - 3$

$x = - \frac{5}{19} - \frac{8 y}{19}$

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 $x = - \frac{5}{19} - \frac{8 y}{19}$ with $- 3$ .

$x = - \frac{5}{19} - \frac{8 (- 3)}{19}$

$y = - 3$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $- \frac{5}{19} - \frac{8 (- 3)}{19}$ .

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Step 4.2.1.1

Combine the numerators over the common denominator.

$x = \frac{- 5 - 8 \cdot - 3}{19}$

$y = - 3$

Step 4.2.1.2

Simplify the expression.

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Step 4.2.1.2.1

Multiply $- 8$ by $- 3$ .

$x = \frac{- 5 + 24}{19}$

$y = - 3$

Step 4.2.1.2.2

Add $- 5$ and $24$ .

$x = \frac{19}{19}$

$y = - 3$

Step 4.2.1.2.3

Divide $19$ by $19$ .

$x = 1$

$y = - 3$

$x = 1$

$y = - 3$

$x = 1$

$y = - 3$

$x = 1$

$y = - 3$

$x = 1$

$y = - 3$

Step 5

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

$(1, - 3)$

Step 6

The result can be shown in multiple forms.

Point Form:

$(1, - 3)$

Equation Form:

$x = 1, y = - 3$

Step 7