Finite Math Examples

$6 x - 7 y = 20$ , $3 x + 5 y = - 7$

Step 1

Solve for $x$ in $6 x - 7 y = 20$ .

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Step 1.1

Add $7 y$ to both sides of the equation.

$6 x = 20 + 7 y$

$3 x + 5 y = - 7$

Step 1.2

Divide each term in $6 x = 20 + 7 y$ by $6$ and simplify.

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Step 1.2.1

Divide each term in $6 x = 20 + 7 y$ by $6$ .

$\frac{6 x}{6} = \frac{20}{6} + \frac{7 y}{6}$

$3 x + 5 y = - 7$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $6$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{6 x}{6} = \frac{20}{6} + \frac{7 y}{6}$

$3 x + 5 y = - 7$

Step 1.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{20}{6} + \frac{7 y}{6}$

$3 x + 5 y = - 7$

$x = \frac{20}{6} + \frac{7 y}{6}$

$3 x + 5 y = - 7$

$x = \frac{20}{6} + \frac{7 y}{6}$

$3 x + 5 y = - 7$

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

Cancel the common factor of $20$ and $6$ .

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Step 1.2.3.1.1

Factor $2$ out of $20$ .

$x = \frac{2 (10)}{6} + \frac{7 y}{6}$

$3 x + 5 y = - 7$

Step 1.2.3.1.2

Cancel the common factors.

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Step 1.2.3.1.2.1

Factor $2$ out of $6$ .

$x = \frac{2 \cdot 10}{2 \cdot 3} + \frac{7 y}{6}$

$3 x + 5 y = - 7$

Step 1.2.3.1.2.2

Cancel the common factor.

$x = \frac{2 \cdot 10}{2 \cdot 3} + \frac{7 y}{6}$

$3 x + 5 y = - 7$

Step 1.2.3.1.2.3

Rewrite the expression.

$x = \frac{10}{3} + \frac{7 y}{6}$

$3 x + 5 y = - 7$

$x = \frac{10}{3} + \frac{7 y}{6}$

$3 x + 5 y = - 7$

$x = \frac{10}{3} + \frac{7 y}{6}$

$3 x + 5 y = - 7$

$x = \frac{10}{3} + \frac{7 y}{6}$

$3 x + 5 y = - 7$

$x = \frac{10}{3} + \frac{7 y}{6}$

$3 x + 5 y = - 7$

$x = \frac{10}{3} + \frac{7 y}{6}$

$3 x + 5 y = - 7$

Step 2

Replace all occurrences of $x$ with $\frac{10}{3} + \frac{7 y}{6}$ in each equation.

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Step 2.1

Replace all occurrences of $x$ in $3 x + 5 y = - 7$ with $\frac{10}{3} + \frac{7 y}{6}$ .

$3 (\frac{10}{3} + \frac{7 y}{6}) + 5 y = - 7$

$x = \frac{10}{3} + \frac{7 y}{6}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $3 (\frac{10}{3} + \frac{7 y}{6}) + 5 y$ .

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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.

$3 (\frac{10}{3}) + 3 (\frac{7 y}{6}) + 5 y = - 7$

$x = \frac{10}{3} + \frac{7 y}{6}$

Step 2.2.1.1.2

Cancel the common factor of $3$ .

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Step 2.2.1.1.2.1

Cancel the common factor.

$3 (\frac{10}{3}) + 3 (\frac{7 y}{6}) + 5 y = - 7$

$x = \frac{10}{3} + \frac{7 y}{6}$

Step 2.2.1.1.2.2

Rewrite the expression.

$10 + 3 (\frac{7 y}{6}) + 5 y = - 7$

$x = \frac{10}{3} + \frac{7 y}{6}$

$10 + 3 (\frac{7 y}{6}) + 5 y = - 7$

$x = \frac{10}{3} + \frac{7 y}{6}$

Step 2.2.1.1.3

Cancel the common factor of $3$ .

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Step 2.2.1.1.3.1

Factor $3$ out of $6$ .

$10 + 3 (\frac{7 y}{3 (2)}) + 5 y = - 7$

$x = \frac{10}{3} + \frac{7 y}{6}$

Step 2.2.1.1.3.2

Cancel the common factor.

$10 + 3 (\frac{7 y}{3 \cdot 2}) + 5 y = - 7$

$x = \frac{10}{3} + \frac{7 y}{6}$

Step 2.2.1.1.3.3

Rewrite the expression.

$10 + \frac{7 y}{2} + 5 y = - 7$

$x = \frac{10}{3} + \frac{7 y}{6}$

$10 + \frac{7 y}{2} + 5 y = - 7$

$x = \frac{10}{3} + \frac{7 y}{6}$

$10 + \frac{7 y}{2} + 5 y = - 7$

$x = \frac{10}{3} + \frac{7 y}{6}$

Step 2.2.1.2

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

$10 + \frac{7 y}{2} + 5 y \cdot \frac{2}{2} = - 7$

$x = \frac{10}{3} + \frac{7 y}{6}$

Step 2.2.1.3

Simplify terms.

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Step 2.2.1.3.1

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

$10 + \frac{7 y}{2} + \frac{5 y \cdot 2}{2} = - 7$

$x = \frac{10}{3} + \frac{7 y}{6}$

Step 2.2.1.3.2

Combine the numerators over the common denominator.

$10 + \frac{7 y + 5 y \cdot 2}{2} = - 7$

$x = \frac{10}{3} + \frac{7 y}{6}$

$10 + \frac{7 y + 5 y \cdot 2}{2} = - 7$

$x = \frac{10}{3} + \frac{7 y}{6}$

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 $7 y + 5 y \cdot 2$ .

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Step 2.2.1.4.1.1.1

Factor $y$ out of $7 y$ .

$10 + \frac{y \cdot 7 + 5 y \cdot 2}{2} = - 7$

$x = \frac{10}{3} + \frac{7 y}{6}$

Step 2.2.1.4.1.1.2

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

$10 + \frac{y \cdot 7 + y (5 \cdot 2)}{2} = - 7$

$x = \frac{10}{3} + \frac{7 y}{6}$

Step 2.2.1.4.1.1.3

Factor $y$ out of $y \cdot 7 + y (5 \cdot 2)$ .

$10 + \frac{y (7 + 5 \cdot 2)}{2} = - 7$

$x = \frac{10}{3} + \frac{7 y}{6}$

$10 + \frac{y (7 + 5 \cdot 2)}{2} = - 7$

$x = \frac{10}{3} + \frac{7 y}{6}$

Step 2.2.1.4.1.2

Multiply $5$ by $2$ .

$10 + \frac{y (7 + 10)}{2} = - 7$

$x = \frac{10}{3} + \frac{7 y}{6}$

Step 2.2.1.4.1.3

Add $7$ and $10$ .

$10 + \frac{y \cdot 17}{2} = - 7$

$x = \frac{10}{3} + \frac{7 y}{6}$

$10 + \frac{y \cdot 17}{2} = - 7$

$x = \frac{10}{3} + \frac{7 y}{6}$

Step 2.2.1.4.2

Move $17$ to the left of $y$ .

$10 + \frac{17 y}{2} = - 7$

$x = \frac{10}{3} + \frac{7 y}{6}$

$10 + \frac{17 y}{2} = - 7$

$x = \frac{10}{3} + \frac{7 y}{6}$

$10 + \frac{17 y}{2} = - 7$

$x = \frac{10}{3} + \frac{7 y}{6}$

$10 + \frac{17 y}{2} = - 7$

$x = \frac{10}{3} + \frac{7 y}{6}$

$10 + \frac{17 y}{2} = - 7$

$x = \frac{10}{3} + \frac{7 y}{6}$

Step 3

Solve for $y$ in $10 + \frac{17 y}{2} = - 7$ .

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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

Subtract $10$ from both sides of the equation.

$\frac{17 y}{2} = - 7 - 10$

$x = \frac{10}{3} + \frac{7 y}{6}$

Step 3.1.2

Subtract $10$ from $- 7$ .

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

$x = \frac{10}{3} + \frac{7 y}{6}$

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

$x = \frac{10}{3} + \frac{7 y}{6}$

Step 3.2

Multiply both sides of the equation by $\frac{2}{17}$ .

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

$x = \frac{10}{3} + \frac{7 y}{6}$

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 $\frac{2}{17} \cdot \frac{17 y}{2}$ .

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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.

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

$x = \frac{10}{3} + \frac{7 y}{6}$

Step 3.3.1.1.1.2

Rewrite the expression.

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

$x = \frac{10}{3} + \frac{7 y}{6}$

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

$x = \frac{10}{3} + \frac{7 y}{6}$

Step 3.3.1.1.2

Cancel the common factor of $17$ .

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Step 3.3.1.1.2.1

Factor $17$ out of $17 y$ .

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

$x = \frac{10}{3} + \frac{7 y}{6}$

Step 3.3.1.1.2.2

Cancel the common factor.

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

$x = \frac{10}{3} + \frac{7 y}{6}$

Step 3.3.1.1.2.3

Rewrite the expression.

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

$x = \frac{10}{3} + \frac{7 y}{6}$

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

$x = \frac{10}{3} + \frac{7 y}{6}$

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

$x = \frac{10}{3} + \frac{7 y}{6}$

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

$x = \frac{10}{3} + \frac{7 y}{6}$

Step 3.3.2

Simplify the right side.

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Step 3.3.2.1

Simplify $\frac{2}{17} \cdot - 17$ .

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Step 3.3.2.1.1

Cancel the common factor of $17$ .

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Step 3.3.2.1.1.1

Factor $17$ out of $- 17$ .

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

$x = \frac{10}{3} + \frac{7 y}{6}$

Step 3.3.2.1.1.2

Cancel the common factor.

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

$x = \frac{10}{3} + \frac{7 y}{6}$

Step 3.3.2.1.1.3

Rewrite the expression.

$y = 2 \cdot - 1$

$x = \frac{10}{3} + \frac{7 y}{6}$

$y = 2 \cdot - 1$

$x = \frac{10}{3} + \frac{7 y}{6}$

Step 3.3.2.1.2

Multiply $2$ by $- 1$ .

$y = - 2$

$x = \frac{10}{3} + \frac{7 y}{6}$

$y = - 2$

$x = \frac{10}{3} + \frac{7 y}{6}$

$y = - 2$

$x = \frac{10}{3} + \frac{7 y}{6}$

$y = - 2$

$x = \frac{10}{3} + \frac{7 y}{6}$

$y = - 2$

$x = \frac{10}{3} + \frac{7 y}{6}$

Step 4

Replace all occurrences of $y$ with $- 2$ in each equation.

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Step 4.1

Replace all occurrences of $y$ in $x = \frac{10}{3} + \frac{7 y}{6}$ with $- 2$ .

$x = \frac{10}{3} + \frac{7 (- 2)}{6}$

$y = - 2$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $\frac{10}{3} + \frac{7 (- 2)}{6}$ .

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Step 4.2.1.1

Multiply $7$ by $- 2$ .

$x = \frac{10}{3} + \frac{- 14}{6}$

$y = - 2$

Step 4.2.1.2

Simplify each term.

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Step 4.2.1.2.1

Cancel the common factor of $- 14$ and $6$ .

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Step 4.2.1.2.1.1

Factor $2$ out of $- 14$ .

$x = \frac{10}{3} + \frac{2 (- 7)}{6}$

$y = - 2$

Step 4.2.1.2.1.2

Cancel the common factors.

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Step 4.2.1.2.1.2.1

Factor $2$ out of $6$ .

$x = \frac{10}{3} + \frac{2 \cdot - 7}{2 \cdot 3}$

$y = - 2$

Step 4.2.1.2.1.2.2

Cancel the common factor.

$x = \frac{10}{3} + \frac{2 \cdot - 7}{2 \cdot 3}$

$y = - 2$

Step 4.2.1.2.1.2.3

Rewrite the expression.

$x = \frac{10}{3} + \frac{- 7}{3}$

$y = - 2$

$x = \frac{10}{3} + \frac{- 7}{3}$

$y = - 2$

$x = \frac{10}{3} + \frac{- 7}{3}$

$y = - 2$

Step 4.2.1.2.2

Move the negative in front of the fraction.

$x = \frac{10}{3} - \frac{7}{3}$

$y = - 2$

$x = \frac{10}{3} - \frac{7}{3}$

$y = - 2$

Step 4.2.1.3

Combine fractions.

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Step 4.2.1.3.1

Combine the numerators over the common denominator.

$x = \frac{10 - 7}{3}$

$y = - 2$

Step 4.2.1.3.2

Simplify the expression.

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Step 4.2.1.3.2.1

Subtract $7$ from $10$ .

$x = \frac{3}{3}$

$y = - 2$

Step 4.2.1.3.2.2

Divide $3$ by $3$ .

$x = 1$

$y = - 2$

$x = 1$

$y = - 2$

$x = 1$

$y = - 2$

$x = 1$

$y = - 2$

$x = 1$

$y = - 2$

$x = 1$

$y = - 2$

Step 5

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

$(1, - 2)$

Step 6

The result can be shown in multiple forms.

Point Form:

$(1, - 2)$

Equation Form:

$x = 1, y = - 2$

Step 7