Finite Math Examples

$3 x - 6 y = 5$ , $4 x - 3 y = 5$

Step 1

Solve for $x$ in $3 x - 6 y = 5$ .

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Step 1.1

Add $6 y$ to both sides of the equation.

$3 x = 5 + 6 y$

$4 x - 3 y = 5$

Step 1.2

Divide each term in $3 x = 5 + 6 y$ by $3$ and simplify.

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Step 1.2.1

Divide each term in $3 x = 5 + 6 y$ by $3$ .

$\frac{3 x}{3} = \frac{5}{3} + \frac{6 y}{3}$

$4 x - 3 y = 5$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $3$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{5}{3} + \frac{6 y}{3}$

$4 x - 3 y = 5$

Step 1.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{5}{3} + \frac{6 y}{3}$

$4 x - 3 y = 5$

$x = \frac{5}{3} + \frac{6 y}{3}$

$4 x - 3 y = 5$

$x = \frac{5}{3} + \frac{6 y}{3}$

$4 x - 3 y = 5$

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

Cancel the common factor of $6$ and $3$ .

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Step 1.2.3.1.1

Factor $3$ out of $6 y$ .

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

$4 x - 3 y = 5$

Step 1.2.3.1.2

Cancel the common factors.

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Step 1.2.3.1.2.1

Factor $3$ out of $3$ .

$x = \frac{5}{3} + \frac{3 (2 y)}{3 (1)}$

$4 x - 3 y = 5$

Step 1.2.3.1.2.2

Cancel the common factor.

$x = \frac{5}{3} + \frac{3 (2 y)}{3 \cdot 1}$

$4 x - 3 y = 5$

Step 1.2.3.1.2.3

Rewrite the expression.

$x = \frac{5}{3} + \frac{2 y}{1}$

$4 x - 3 y = 5$

Step 1.2.3.1.2.4

Divide $2 y$ by $1$ .

$x = \frac{5}{3} + 2 y$

$4 x - 3 y = 5$

$x = \frac{5}{3} + 2 y$

$4 x - 3 y = 5$

$x = \frac{5}{3} + 2 y$

$4 x - 3 y = 5$

$x = \frac{5}{3} + 2 y$

$4 x - 3 y = 5$

$x = \frac{5}{3} + 2 y$

$4 x - 3 y = 5$

$x = \frac{5}{3} + 2 y$

$4 x - 3 y = 5$

Step 2

Replace all occurrences of $x$ with $\frac{5}{3} + 2 y$ in each equation.

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Step 2.1

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

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

$x = \frac{5}{3} + 2 y$

Step 2.2

Simplify the left side.

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Step 2.2.1

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

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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.

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

$x = \frac{5}{3} + 2 y$

Step 2.2.1.1.2

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

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Step 2.2.1.1.2.1

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

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

$x = \frac{5}{3} + 2 y$

Step 2.2.1.1.2.2

Multiply $4$ by $5$ .

$\frac{20}{3} + 4 (2 y) - 3 y = 5$

$x = \frac{5}{3} + 2 y$

$\frac{20}{3} + 4 (2 y) - 3 y = 5$

$x = \frac{5}{3} + 2 y$

Step 2.2.1.1.3

Multiply $2$ by $4$ .

$\frac{20}{3} + 8 y - 3 y = 5$

$x = \frac{5}{3} + 2 y$

$\frac{20}{3} + 8 y - 3 y = 5$

$x = \frac{5}{3} + 2 y$

Step 2.2.1.2

Subtract $3 y$ from $8 y$ .

$\frac{20}{3} + 5 y = 5$

$x = \frac{5}{3} + 2 y$

$\frac{20}{3} + 5 y = 5$

$x = \frac{5}{3} + 2 y$

$\frac{20}{3} + 5 y = 5$

$x = \frac{5}{3} + 2 y$

$\frac{20}{3} + 5 y = 5$

$x = \frac{5}{3} + 2 y$

Step 3

Solve for $y$ in $\frac{20}{3} + 5 y = 5$ .

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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 $\frac{20}{3}$ from both sides of the equation.

$5 y = 5 - \frac{20}{3}$

$x = \frac{5}{3} + 2 y$

Step 3.1.2

To write $5$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$5 y = 5 \cdot \frac{3}{3} - \frac{20}{3}$

$x = \frac{5}{3} + 2 y$

Step 3.1.3

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

$5 y = \frac{5 \cdot 3}{3} - \frac{20}{3}$

$x = \frac{5}{3} + 2 y$

Step 3.1.4

Combine the numerators over the common denominator.

$5 y = \frac{5 \cdot 3 - 20}{3}$

$x = \frac{5}{3} + 2 y$

Step 3.1.5

Simplify the numerator.

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Step 3.1.5.1

Multiply $5$ by $3$ .

$5 y = \frac{15 - 20}{3}$

$x = \frac{5}{3} + 2 y$

Step 3.1.5.2

Subtract $20$ from $15$ .

$5 y = \frac{- 5}{3}$

$x = \frac{5}{3} + 2 y$

$5 y = \frac{- 5}{3}$

$x = \frac{5}{3} + 2 y$

Step 3.1.6

Move the negative in front of the fraction.

$5 y = - \frac{5}{3}$

$x = \frac{5}{3} + 2 y$

$5 y = - \frac{5}{3}$

$x = \frac{5}{3} + 2 y$

Step 3.2

Divide each term in $5 y = - \frac{5}{3}$ by $5$ and simplify.

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Step 3.2.1

Divide each term in $5 y = - \frac{5}{3}$ by $5$ .

$\frac{5 y}{5} = \frac{- \frac{5}{3}}{5}$

$x = \frac{5}{3} + 2 y$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $5$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{5 y}{5} = \frac{- \frac{5}{3}}{5}$

$x = \frac{5}{3} + 2 y$

Step 3.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{- \frac{5}{3}}{5}$

$x = \frac{5}{3} + 2 y$

$y = \frac{- \frac{5}{3}}{5}$

$x = \frac{5}{3} + 2 y$

$y = \frac{- \frac{5}{3}}{5}$

$x = \frac{5}{3} + 2 y$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Multiply the numerator by the reciprocal of the denominator.

$y = - \frac{5}{3} \cdot \frac{1}{5}$

$x = \frac{5}{3} + 2 y$

Step 3.2.3.2

Cancel the common factor of $5$ .

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Step 3.2.3.2.1

Move the leading negative in $- \frac{5}{3}$ into the numerator.

$y = \frac{- 5}{3} \cdot \frac{1}{5}$

$x = \frac{5}{3} + 2 y$

Step 3.2.3.2.2

Factor $5$ out of $- 5$ .

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

$x = \frac{5}{3} + 2 y$

Step 3.2.3.2.3

Cancel the common factor.

$y = \frac{5 \cdot - 1}{3} \cdot \frac{1}{5}$

$x = \frac{5}{3} + 2 y$

Step 3.2.3.2.4

Rewrite the expression.

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

$x = \frac{5}{3} + 2 y$

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

$x = \frac{5}{3} + 2 y$

Step 3.2.3.3

Move the negative in front of the fraction.

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

$x = \frac{5}{3} + 2 y$

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

$x = \frac{5}{3} + 2 y$

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

$x = \frac{5}{3} + 2 y$

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

$x = \frac{5}{3} + 2 y$

Step 4

Replace all occurrences of $y$ with $- \frac{1}{3}$ in each equation.

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Step 4.1

Replace all occurrences of $y$ in $x = \frac{5}{3} + 2 y$ with $- \frac{1}{3}$ .

$x = \frac{5}{3} + 2 (- \frac{1}{3})$

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

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $\frac{5}{3} + 2 (- \frac{1}{3})$ .

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Step 4.2.1.1

Simplify each term.

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Step 4.2.1.1.1

Multiply $2 (- \frac{1}{3})$ .

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Step 4.2.1.1.1.1

Multiply $- 1$ by $2$ .

$x = \frac{5}{3} - 2 (\frac{1}{3})$

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

Step 4.2.1.1.1.2

Combine $- 2$ and $\frac{1}{3}$ .

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

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

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

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

Step 4.2.1.1.2

Move the negative in front of the fraction.

$x = \frac{5}{3} - \frac{2}{3}$

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

$x = \frac{5}{3} - \frac{2}{3}$

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

Step 4.2.1.2

Combine fractions.

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Step 4.2.1.2.1

Combine the numerators over the common denominator.

$x = \frac{5 - 2}{3}$

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

Step 4.2.1.2.2

Simplify the expression.

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Step 4.2.1.2.2.1

Subtract $2$ from $5$ .

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

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

Step 4.2.1.2.2.2

Divide $3$ by $3$ .

$x = 1$

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

$x = 1$

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

$x = 1$

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

$x = 1$

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

$x = 1$

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

$x = 1$

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

Step 5

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

$(1, - \frac{1}{3})$

Step 6

The result can be shown in multiple forms.

Point Form:

$(1, - \frac{1}{3})$

Equation Form:

$x = 1, y = - \frac{1}{3}$

Step 7