Finite Math Examples

$\frac{7}{2} x + \frac{3}{2} y = 0$ , $\frac{1}{3} x - \frac{20}{7} y = 4$

Step 1

Solve for $x$ in $\frac{7}{2} x + \frac{3}{2} y = 0$ .

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Step 1.1

Simplify each term.

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Step 1.1.1

Combine $\frac{7}{2}$ and $x$ .

$\frac{7 x}{2} + \frac{3}{2} y = 0$

$\frac{1}{3} x - \frac{20}{7} y = 4$

Step 1.1.2

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

$\frac{7 x}{2} + \frac{3 y}{2} = 0$

$\frac{1}{3} x - \frac{20}{7} y = 4$

$\frac{7 x}{2} + \frac{3 y}{2} = 0$

$\frac{1}{3} x - \frac{20}{7} y = 4$

Step 1.2

Subtract $\frac{3 y}{2}$ from both sides of the equation.

$\frac{7 x}{2} = - \frac{3 y}{2}$

$\frac{1}{3} x - \frac{20}{7} y = 4$

Step 1.3

Since the expression on each side of the equation has the same denominator, the numerators must be equal.

$7 x = - 3 y$

$\frac{1}{3} x - \frac{20}{7} y = 4$

Step 1.4

Divide each term in $7 x = - 3 y$ by $7$ and simplify.

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Step 1.4.1

Divide each term in $7 x = - 3 y$ by $7$ .

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

$\frac{1}{3} x - \frac{20}{7} y = 4$

Step 1.4.2

Simplify the left side.

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Step 1.4.2.1

Cancel the common factor of $7$ .

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Step 1.4.2.1.1

Cancel the common factor.

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

$\frac{1}{3} x - \frac{20}{7} y = 4$

Step 1.4.2.1.2

Divide $x$ by $1$ .

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

$\frac{1}{3} x - \frac{20}{7} y = 4$

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

$\frac{1}{3} x - \frac{20}{7} y = 4$

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

$\frac{1}{3} x - \frac{20}{7} y = 4$

Step 1.4.3

Simplify the right side.

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Step 1.4.3.1

Move the negative in front of the fraction.

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

$\frac{1}{3} x - \frac{20}{7} y = 4$

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

$\frac{1}{3} x - \frac{20}{7} y = 4$

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

$\frac{1}{3} x - \frac{20}{7} y = 4$

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

$\frac{1}{3} x - \frac{20}{7} y = 4$

Step 2

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

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Step 2.1

Replace all occurrences of $x$ in $\frac{1}{3} x - \frac{20}{7} y = 4$ with $- \frac{3 y}{7}$ .

$\frac{1}{3} \cdot (- \frac{3 y}{7}) - \frac{20}{7} y = 4$

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

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $\frac{1}{3} \cdot (- \frac{3 y}{7}) - \frac{20}{7} y$ .

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Step 2.2.1.1

Simplify each term.

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Step 2.2.1.1.1

Cancel the common factor of $3$ .

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Step 2.2.1.1.1.1

Move the leading negative in $- \frac{3 y}{7}$ into the numerator.

$\frac{1}{3} \cdot \frac{- 3 y}{7} - \frac{20}{7} y = 4$

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

Step 2.2.1.1.1.2

Factor $3$ out of $- 3 y$ .

$\frac{1}{3} \cdot \frac{3 (- y)}{7} - \frac{20}{7} y = 4$

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

Step 2.2.1.1.1.3

Cancel the common factor.

$\frac{1}{3} \cdot \frac{3 (- y)}{7} - \frac{20}{7} y = 4$

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

Step 2.2.1.1.1.4

Rewrite the expression.

$\frac{- y}{7} - \frac{20}{7} y = 4$

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

$\frac{- y}{7} - \frac{20}{7} y = 4$

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

Step 2.2.1.1.2

Move the negative in front of the fraction.

$- \frac{y}{7} - \frac{20}{7} y = 4$

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

Step 2.2.1.1.3

Combine $y$ and $\frac{20}{7}$ .

$- \frac{y}{7} - \frac{y \cdot 20}{7} = 4$

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

Step 2.2.1.1.4

Move $20$ to the left of $y$ .

$- \frac{y}{7} - \frac{20 y}{7} = 4$

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

$- \frac{y}{7} - \frac{20 y}{7} = 4$

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

Step 2.2.1.2

Simplify terms.

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Step 2.2.1.2.1

Combine the numerators over the common denominator.

$\frac{- y - 20 y}{7} = 4$

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

Step 2.2.1.2.2

Subtract $20 y$ from $- y$ .

$\frac{- 21 y}{7} = 4$

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

Step 2.2.1.2.3

Cancel the common factor of $- 21$ and $7$ .

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Step 2.2.1.2.3.1

Factor $7$ out of $- 21 y$ .

$\frac{7 (- 3 y)}{7} = 4$

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

Step 2.2.1.2.3.2

Cancel the common factors.

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Step 2.2.1.2.3.2.1

Factor $7$ out of $7$ .

$\frac{7 (- 3 y)}{7 (1)} = 4$

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

Step 2.2.1.2.3.2.2

Cancel the common factor.

$\frac{7 (- 3 y)}{7 \cdot 1} = 4$

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

Step 2.2.1.2.3.2.3

Rewrite the expression.

$\frac{- 3 y}{1} = 4$

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

Step 2.2.1.2.3.2.4

Divide $- 3 y$ by $1$ .

$- 3 y = 4$

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

$- 3 y = 4$

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

$- 3 y = 4$

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

$- 3 y = 4$

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

$- 3 y = 4$

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

$- 3 y = 4$

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

$- 3 y = 4$

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

Step 3

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

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Step 3.1

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

$\frac{- 3 y}{- 3} = \frac{4}{- 3}$

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

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $- 3$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{- 3 y}{- 3} = \frac{4}{- 3}$

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

Step 3.2.1.2

Divide $y$ by $1$ .

$y = \frac{4}{- 3}$

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

$y = \frac{4}{- 3}$

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

$y = \frac{4}{- 3}$

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

Step 3.3

Simplify the right side.

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Step 3.3.1

Move the negative in front of the fraction.

$y = - \frac{4}{3}$

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

$y = - \frac{4}{3}$

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

$y = - \frac{4}{3}$

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

Step 4

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

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Step 4.1

Replace all occurrences of $y$ in $x = - \frac{3 y}{7}$ with $- \frac{4}{3}$ .

$x = - \frac{3 (- \frac{4}{3})}{7}$

$y = - \frac{4}{3}$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $- \frac{3 (- \frac{4}{3})}{7}$ .

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Step 4.2.1.1

Simplify the numerator.

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Step 4.2.1.1.1

Multiply $- 1$ by $3$ .

$x = - \frac{- 3 (\frac{4}{3})}{7}$

$y = - \frac{4}{3}$

Step 4.2.1.1.2

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

$x = - \frac{\frac{- 3 \cdot 4}{3}}{7}$

$y = - \frac{4}{3}$

$x = - \frac{\frac{- 3 \cdot 4}{3}}{7}$

$y = - \frac{4}{3}$

Step 4.2.1.2

Simplify the expression.

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Step 4.2.1.2.1

Multiply $- 3$ by $4$ .

$x = - \frac{\frac{- 12}{3}}{7}$

$y = - \frac{4}{3}$

Step 4.2.1.2.2

Divide $- 12$ by $3$ .

$x = - \frac{- 4}{7}$

$y = - \frac{4}{3}$

Step 4.2.1.2.3

Move the negative in front of the fraction.

$x = \frac{4}{7}$

$y = - \frac{4}{3}$

$x = \frac{4}{7}$

$y = - \frac{4}{3}$

Step 4.2.1.3

Multiply $- - \frac{4}{7}$ .

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Step 4.2.1.3.1

Multiply $- 1$ by $- 1$ .

$x = 1 (\frac{4}{7})$

$y = - \frac{4}{3}$

Step 4.2.1.3.2

Multiply $\frac{4}{7}$ by $1$ .

$x = \frac{4}{7}$

$y = - \frac{4}{3}$

$x = \frac{4}{7}$

$y = - \frac{4}{3}$

$x = \frac{4}{7}$

$y = - \frac{4}{3}$

$x = \frac{4}{7}$

$y = - \frac{4}{3}$

$x = \frac{4}{7}$

$y = - \frac{4}{3}$

Step 5

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

$(\frac{4}{7}, - \frac{4}{3})$

Step 6

The result can be shown in multiple forms.

Point Form:

$(\frac{4}{7}, - \frac{4}{3})$

Equation Form:

$x = \frac{4}{7}, y = - \frac{4}{3}$

Step 7