Finite Math Examples

$4 x - 3 y = - 22$ , $2 x + 5 y = 6$

Step 1

Solve for $x$ in $4 x - 3 y = - 22$ .

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Step 1.1

Add $3 y$ to both sides of the equation.

$4 x = - 22 + 3 y$

$2 x + 5 y = 6$

Step 1.2

Divide each term in $4 x = - 22 + 3 y$ by $4$ and simplify.

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Step 1.2.1

Divide each term in $4 x = - 22 + 3 y$ by $4$ .

$\frac{4 x}{4} = \frac{- 22}{4} + \frac{3 y}{4}$

$2 x + 5 y = 6$

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.

$\frac{4 x}{4} = \frac{- 22}{4} + \frac{3 y}{4}$

$2 x + 5 y = 6$

Step 1.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 22}{4} + \frac{3 y}{4}$

$2 x + 5 y = 6$

$x = \frac{- 22}{4} + \frac{3 y}{4}$

$2 x + 5 y = 6$

$x = \frac{- 22}{4} + \frac{3 y}{4}$

$2 x + 5 y = 6$

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

Cancel the common factor of $- 22$ and $4$ .

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Step 1.2.3.1.1.1

Factor $2$ out of $- 22$ .

$x = \frac{2 (- 11)}{4} + \frac{3 y}{4}$

$2 x + 5 y = 6$

Step 1.2.3.1.1.2

Cancel the common factors.

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Step 1.2.3.1.1.2.1

Factor $2$ out of $4$ .

$x = \frac{2 \cdot - 11}{2 \cdot 2} + \frac{3 y}{4}$

$2 x + 5 y = 6$

Step 1.2.3.1.1.2.2

Cancel the common factor.

$x = \frac{2 \cdot - 11}{2 \cdot 2} + \frac{3 y}{4}$

$2 x + 5 y = 6$

Step 1.2.3.1.1.2.3

Rewrite the expression.

$x = \frac{- 11}{2} + \frac{3 y}{4}$

$2 x + 5 y = 6$

$x = \frac{- 11}{2} + \frac{3 y}{4}$

$2 x + 5 y = 6$

$x = \frac{- 11}{2} + \frac{3 y}{4}$

$2 x + 5 y = 6$

Step 1.2.3.1.2

Move the negative in front of the fraction.

$x = - \frac{11}{2} + \frac{3 y}{4}$

$2 x + 5 y = 6$

$x = - \frac{11}{2} + \frac{3 y}{4}$

$2 x + 5 y = 6$

$x = - \frac{11}{2} + \frac{3 y}{4}$

$2 x + 5 y = 6$

$x = - \frac{11}{2} + \frac{3 y}{4}$

$2 x + 5 y = 6$

$x = - \frac{11}{2} + \frac{3 y}{4}$

$2 x + 5 y = 6$

Step 2

Replace all occurrences of $x$ with $- \frac{11}{2} + \frac{3 y}{4}$ in each equation.

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Step 2.1

Replace all occurrences of $x$ in $2 x + 5 y = 6$ with $- \frac{11}{2} + \frac{3 y}{4}$ .

$2 (- \frac{11}{2} + \frac{3 y}{4}) + 5 y = 6$

$x = - \frac{11}{2} + \frac{3 y}{4}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $2 (- \frac{11}{2} + \frac{3 y}{4}) + 5 y$ .

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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.

$2 (- \frac{11}{2}) + 2 (\frac{3 y}{4}) + 5 y = 6$

$x = - \frac{11}{2} + \frac{3 y}{4}$

Step 2.2.1.1.2

Cancel the common factor of $2$ .

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Step 2.2.1.1.2.1

Move the leading negative in $- \frac{11}{2}$ into the numerator.

$2 (\frac{- 11}{2}) + 2 (\frac{3 y}{4}) + 5 y = 6$

$x = - \frac{11}{2} + \frac{3 y}{4}$

Step 2.2.1.1.2.2

Cancel the common factor.

$2 (\frac{- 11}{2}) + 2 (\frac{3 y}{4}) + 5 y = 6$

$x = - \frac{11}{2} + \frac{3 y}{4}$

Step 2.2.1.1.2.3

Rewrite the expression.

$- 11 + 2 (\frac{3 y}{4}) + 5 y = 6$

$x = - \frac{11}{2} + \frac{3 y}{4}$

$- 11 + 2 (\frac{3 y}{4}) + 5 y = 6$

$x = - \frac{11}{2} + \frac{3 y}{4}$

Step 2.2.1.1.3

Cancel the common factor of $2$ .

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Step 2.2.1.1.3.1

Factor $2$ out of $4$ .

$- 11 + 2 (\frac{3 y}{2 (2)}) + 5 y = 6$

$x = - \frac{11}{2} + \frac{3 y}{4}$

Step 2.2.1.1.3.2

Cancel the common factor.

$- 11 + 2 (\frac{3 y}{2 \cdot 2}) + 5 y = 6$

$x = - \frac{11}{2} + \frac{3 y}{4}$

Step 2.2.1.1.3.3

Rewrite the expression.

$- 11 + \frac{3 y}{2} + 5 y = 6$

$x = - \frac{11}{2} + \frac{3 y}{4}$

$- 11 + \frac{3 y}{2} + 5 y = 6$

$x = - \frac{11}{2} + \frac{3 y}{4}$

$- 11 + \frac{3 y}{2} + 5 y = 6$

$x = - \frac{11}{2} + \frac{3 y}{4}$

Step 2.2.1.2

To write $5 y$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$- 11 + \frac{3 y}{2} + 5 y \cdot \frac{2}{2} = 6$

$x = - \frac{11}{2} + \frac{3 y}{4}$

Step 2.2.1.3

Simplify terms.

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Step 2.2.1.3.1

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

$- 11 + \frac{3 y}{2} + \frac{5 y \cdot 2}{2} = 6$

$x = - \frac{11}{2} + \frac{3 y}{4}$

Step 2.2.1.3.2

Combine the numerators over the common denominator.

$- 11 + \frac{3 y + 5 y \cdot 2}{2} = 6$

$x = - \frac{11}{2} + \frac{3 y}{4}$

$- 11 + \frac{3 y + 5 y \cdot 2}{2} = 6$

$x = - \frac{11}{2} + \frac{3 y}{4}$

Step 2.2.1.4

Simplify each term.

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Step 2.2.1.4.1

Simplify the numerator.

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Step 2.2.1.4.1.1

Factor $y$ out of $3 y + 5 y \cdot 2$ .

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Step 2.2.1.4.1.1.1

Factor $y$ out of $3 y$ .

$- 11 + \frac{y \cdot 3 + 5 y \cdot 2}{2} = 6$

$x = - \frac{11}{2} + \frac{3 y}{4}$

Step 2.2.1.4.1.1.2

Factor $y$ out of $5 y \cdot 2$ .

$- 11 + \frac{y \cdot 3 + y (5 \cdot 2)}{2} = 6$

$x = - \frac{11}{2} + \frac{3 y}{4}$

Step 2.2.1.4.1.1.3

Factor $y$ out of $y \cdot 3 + y (5 \cdot 2)$ .

$- 11 + \frac{y (3 + 5 \cdot 2)}{2} = 6$

$x = - \frac{11}{2} + \frac{3 y}{4}$

$- 11 + \frac{y (3 + 5 \cdot 2)}{2} = 6$

$x = - \frac{11}{2} + \frac{3 y}{4}$

Step 2.2.1.4.1.2

Multiply $5$ by $2$ .

$- 11 + \frac{y (3 + 10)}{2} = 6$

$x = - \frac{11}{2} + \frac{3 y}{4}$

Step 2.2.1.4.1.3

Add $3$ and $10$ .

$- 11 + \frac{y \cdot 13}{2} = 6$

$x = - \frac{11}{2} + \frac{3 y}{4}$

$- 11 + \frac{y \cdot 13}{2} = 6$

$x = - \frac{11}{2} + \frac{3 y}{4}$

Step 2.2.1.4.2

Move $13$ to the left of $y$ .

$- 11 + \frac{13 y}{2} = 6$

$x = - \frac{11}{2} + \frac{3 y}{4}$

$- 11 + \frac{13 y}{2} = 6$

$x = - \frac{11}{2} + \frac{3 y}{4}$

$- 11 + \frac{13 y}{2} = 6$

$x = - \frac{11}{2} + \frac{3 y}{4}$

$- 11 + \frac{13 y}{2} = 6$

$x = - \frac{11}{2} + \frac{3 y}{4}$

$- 11 + \frac{13 y}{2} = 6$

$x = - \frac{11}{2} + \frac{3 y}{4}$

Step 3

Solve for $y$ in $- 11 + \frac{13 y}{2} = 6$ .

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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 $11$ to both sides of the equation.

$\frac{13 y}{2} = 6 + 11$

$x = - \frac{11}{2} + \frac{3 y}{4}$

Step 3.1.2

Add $6$ and $11$ .

$\frac{13 y}{2} = 17$

$x = - \frac{11}{2} + \frac{3 y}{4}$

$\frac{13 y}{2} = 17$

$x = - \frac{11}{2} + \frac{3 y}{4}$

Step 3.2

Multiply both sides of the equation by $\frac{2}{13}$ .

$\frac{2}{13} \cdot \frac{13 y}{2} = \frac{2}{13} \cdot 17$

$x = - \frac{11}{2} + \frac{3 y}{4}$

Step 3.3

Simplify both sides of the equation.

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Step 3.3.1

Simplify the left side.

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Step 3.3.1.1

Simplify $\frac{2}{13} \cdot \frac{13 y}{2}$ .

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Step 3.3.1.1.1

Cancel the common factor of $2$ .

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Step 3.3.1.1.1.1

Cancel the common factor.

$\frac{2}{13} \cdot \frac{13 y}{2} = \frac{2}{13} \cdot 17$

$x = - \frac{11}{2} + \frac{3 y}{4}$

Step 3.3.1.1.1.2

Rewrite the expression.

$\frac{1}{13} \cdot (13 y) = \frac{2}{13} \cdot 17$

$x = - \frac{11}{2} + \frac{3 y}{4}$

$\frac{1}{13} \cdot (13 y) = \frac{2}{13} \cdot 17$

$x = - \frac{11}{2} + \frac{3 y}{4}$

Step 3.3.1.1.2

Cancel the common factor of $13$ .

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Step 3.3.1.1.2.1

Factor $13$ out of $13 y$ .

$\frac{1}{13} \cdot (13 (y)) = \frac{2}{13} \cdot 17$

$x = - \frac{11}{2} + \frac{3 y}{4}$

Step 3.3.1.1.2.2

Cancel the common factor.

$\frac{1}{13} \cdot (13 y) = \frac{2}{13} \cdot 17$

$x = - \frac{11}{2} + \frac{3 y}{4}$

Step 3.3.1.1.2.3

Rewrite the expression.

$y = \frac{2}{13} \cdot 17$

$x = - \frac{11}{2} + \frac{3 y}{4}$

$y = \frac{2}{13} \cdot 17$

$x = - \frac{11}{2} + \frac{3 y}{4}$

$y = \frac{2}{13} \cdot 17$

$x = - \frac{11}{2} + \frac{3 y}{4}$

$y = \frac{2}{13} \cdot 17$

$x = - \frac{11}{2} + \frac{3 y}{4}$

Step 3.3.2

Simplify the right side.

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Step 3.3.2.1

Multiply $\frac{2}{13} \cdot 17$ .

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Step 3.3.2.1.1

Combine $\frac{2}{13}$ and $17$ .

$y = \frac{2 \cdot 17}{13}$

$x = - \frac{11}{2} + \frac{3 y}{4}$

Step 3.3.2.1.2

Multiply $2$ by $17$ .

$y = \frac{34}{13}$

$x = - \frac{11}{2} + \frac{3 y}{4}$

$y = \frac{34}{13}$

$x = - \frac{11}{2} + \frac{3 y}{4}$

$y = \frac{34}{13}$

$x = - \frac{11}{2} + \frac{3 y}{4}$

$y = \frac{34}{13}$

$x = - \frac{11}{2} + \frac{3 y}{4}$

$y = \frac{34}{13}$

$x = - \frac{11}{2} + \frac{3 y}{4}$

Step 4

Replace all occurrences of $y$ with $\frac{34}{13}$ in each equation.

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Step 4.1

Replace all occurrences of $y$ in $x = - \frac{11}{2} + \frac{3 y}{4}$ with $\frac{34}{13}$ .

$x = - \frac{11}{2} + \frac{3 (\frac{34}{13})}{4}$

$y = \frac{34}{13}$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $- \frac{11}{2} + \frac{3 (\frac{34}{13})}{4}$ .

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Step 4.2.1.1

Simplify each term.

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Step 4.2.1.1.1

Combine $3$ and $\frac{34}{13}$ .

$x = - \frac{11}{2} + \frac{\frac{3 \cdot 34}{13}}{4}$

$y = \frac{34}{13}$

Step 4.2.1.1.2

Multiply $3$ by $34$ .

$x = - \frac{11}{2} + \frac{\frac{102}{13}}{4}$

$y = \frac{34}{13}$

Step 4.2.1.1.3

Multiply the numerator by the reciprocal of the denominator.

$x = - \frac{11}{2} + \frac{102}{13} \cdot \frac{1}{4}$

$y = \frac{34}{13}$

Step 4.2.1.1.4

Cancel the common factor of $2$ .

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Step 4.2.1.1.4.1

Factor $2$ out of $102$ .

$x = - \frac{11}{2} + \frac{2 (51)}{13} \cdot \frac{1}{4}$

$y = \frac{34}{13}$

Step 4.2.1.1.4.2

Factor $2$ out of $4$ .

$x = - \frac{11}{2} + \frac{2 \cdot 51}{13} \cdot \frac{1}{2 \cdot 2}$

$y = \frac{34}{13}$

Step 4.2.1.1.4.3

Cancel the common factor.

$x = - \frac{11}{2} + \frac{2 \cdot 51}{13} \cdot \frac{1}{2 \cdot 2}$

$y = \frac{34}{13}$

Step 4.2.1.1.4.4

Rewrite the expression.

$x = - \frac{11}{2} + \frac{51}{13} \cdot \frac{1}{2}$

$y = \frac{34}{13}$

$x = - \frac{11}{2} + \frac{51}{13} \cdot \frac{1}{2}$

$y = \frac{34}{13}$

Step 4.2.1.1.5

Multiply $\frac{51}{13}$ by $\frac{1}{2}$ .

$x = - \frac{11}{2} + \frac{51}{13 \cdot 2}$

$y = \frac{34}{13}$

Step 4.2.1.1.6

Multiply $13$ by $2$ .

$x = - \frac{11}{2} + \frac{51}{26}$

$y = \frac{34}{13}$

$x = - \frac{11}{2} + \frac{51}{26}$

$y = \frac{34}{13}$

Step 4.2.1.2

To write $- \frac{11}{2}$ as a fraction with a common denominator, multiply by $\frac{13}{13}$ .

$x = - \frac{11}{2} \cdot \frac{13}{13} + \frac{51}{26}$

$y = \frac{34}{13}$

Step 4.2.1.3

Write each expression with a common denominator of $26$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.2.1.3.1

Multiply $\frac{11}{2}$ by $\frac{13}{13}$ .

$x = - \frac{11 \cdot 13}{2 \cdot 13} + \frac{51}{26}$

$y = \frac{34}{13}$

Step 4.2.1.3.2

Multiply $2$ by $13$ .

$x = - \frac{11 \cdot 13}{26} + \frac{51}{26}$

$y = \frac{34}{13}$

$x = - \frac{11 \cdot 13}{26} + \frac{51}{26}$

$y = \frac{34}{13}$

Step 4.2.1.4

Combine the numerators over the common denominator.

$x = \frac{- 11 \cdot 13 + 51}{26}$

$y = \frac{34}{13}$

Step 4.2.1.5

Simplify the numerator.

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Step 4.2.1.5.1

Multiply $- 11$ by $13$ .

$x = \frac{- 143 + 51}{26}$

$y = \frac{34}{13}$

Step 4.2.1.5.2

Add $- 143$ and $51$ .

$x = \frac{- 92}{26}$

$y = \frac{34}{13}$

$x = \frac{- 92}{26}$

$y = \frac{34}{13}$

Step 4.2.1.6

Cancel the common factor of $- 92$ and $26$ .

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Step 4.2.1.6.1

Factor $2$ out of $- 92$ .

$x = \frac{2 (- 46)}{26}$

$y = \frac{34}{13}$

Step 4.2.1.6.2

Cancel the common factors.

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Step 4.2.1.6.2.1

Factor $2$ out of $26$ .

$x = \frac{2 \cdot - 46}{2 \cdot 13}$

$y = \frac{34}{13}$

Step 4.2.1.6.2.2

Cancel the common factor.

$x = \frac{2 \cdot - 46}{2 \cdot 13}$

$y = \frac{34}{13}$

Step 4.2.1.6.2.3

Rewrite the expression.

$x = \frac{- 46}{13}$

$y = \frac{34}{13}$

$x = \frac{- 46}{13}$

$y = \frac{34}{13}$

$x = \frac{- 46}{13}$

$y = \frac{34}{13}$

Step 4.2.1.7

Move the negative in front of the fraction.

$x = - \frac{46}{13}$

$y = \frac{34}{13}$

$x = - \frac{46}{13}$

$y = \frac{34}{13}$

$x = - \frac{46}{13}$

$y = \frac{34}{13}$

$x = - \frac{46}{13}$

$y = \frac{34}{13}$

Step 5

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

$(- \frac{46}{13}, \frac{34}{13})$

Step 6

The result can be shown in multiple forms.

Point Form:

$(- \frac{46}{13}, \frac{34}{13})$

Equation Form:

$x = - \frac{46}{13}, y = \frac{34}{13}$

Step 7