Finite Math Examples

$6 x - y - 16 = 0$ , $5 x + 2 y - 2 = 0$

Step 1

Solve for $x$ in $6 x - y - 16 = 0$ .

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Step 1.1

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

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Step 1.1.1

Add $y$ to both sides of the equation.

$6 x - 16 = y$

$5 x + 2 y - 2 = 0$

Step 1.1.2

Add $16$ to both sides of the equation.

$6 x = y + 16$

$5 x + 2 y - 2 = 0$

$6 x = y + 16$

$5 x + 2 y - 2 = 0$

Step 1.2

Divide each term in $6 x = y + 16$ by $6$ and simplify.

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Step 1.2.1

Divide each term in $6 x = y + 16$ by $6$ .

$\frac{6 x}{6} = \frac{y}{6} + \frac{16}{6}$

$5 x + 2 y - 2 = 0$

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.

$\frac{6 x}{6} = \frac{y}{6} + \frac{16}{6}$

$5 x + 2 y - 2 = 0$

Step 1.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{y}{6} + \frac{16}{6}$

$5 x + 2 y - 2 = 0$

$x = \frac{y}{6} + \frac{16}{6}$

$5 x + 2 y - 2 = 0$

$x = \frac{y}{6} + \frac{16}{6}$

$5 x + 2 y - 2 = 0$

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

Cancel the common factor of $16$ and $6$ .

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Step 1.2.3.1.1

Factor $2$ out of $16$ .

$x = \frac{y}{6} + \frac{2 (8)}{6}$

$5 x + 2 y - 2 = 0$

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 $6$ .

$x = \frac{y}{6} + \frac{2 \cdot 8}{2 \cdot 3}$

$5 x + 2 y - 2 = 0$

Step 1.2.3.1.2.2

Cancel the common factor.

$x = \frac{y}{6} + \frac{2 \cdot 8}{2 \cdot 3}$

$5 x + 2 y - 2 = 0$

Step 1.2.3.1.2.3

Rewrite the expression.

$x = \frac{y}{6} + \frac{8}{3}$

$5 x + 2 y - 2 = 0$

$x = \frac{y}{6} + \frac{8}{3}$

$5 x + 2 y - 2 = 0$

$x = \frac{y}{6} + \frac{8}{3}$

$5 x + 2 y - 2 = 0$

$x = \frac{y}{6} + \frac{8}{3}$

$5 x + 2 y - 2 = 0$

$x = \frac{y}{6} + \frac{8}{3}$

$5 x + 2 y - 2 = 0$

$x = \frac{y}{6} + \frac{8}{3}$

$5 x + 2 y - 2 = 0$

Step 2

Replace all occurrences of $x$ with $\frac{y}{6} + \frac{8}{3}$ in each equation.

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Step 2.1

Replace all occurrences of $x$ in $5 x + 2 y - 2 = 0$ with $\frac{y}{6} + \frac{8}{3}$ .

$5 (\frac{y}{6} + \frac{8}{3}) + 2 y - 2 = 0$

$x = \frac{y}{6} + \frac{8}{3}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $5 (\frac{y}{6} + \frac{8}{3}) + 2 y - 2$ .

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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.

$5 (\frac{y}{6}) + 5 (\frac{8}{3}) + 2 y - 2 = 0$

$x = \frac{y}{6} + \frac{8}{3}$

Step 2.2.1.1.2

Combine $5$ and $\frac{y}{6}$ .

$\frac{5 y}{6} + 5 (\frac{8}{3}) + 2 y - 2 = 0$

$x = \frac{y}{6} + \frac{8}{3}$

Step 2.2.1.1.3

Multiply $5 (\frac{8}{3})$ .

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Step 2.2.1.1.3.1

Combine $5$ and $\frac{8}{3}$ .

$\frac{5 y}{6} + \frac{5 \cdot 8}{3} + 2 y - 2 = 0$

$x = \frac{y}{6} + \frac{8}{3}$

Step 2.2.1.1.3.2

Multiply $5$ by $8$ .

$\frac{5 y}{6} + \frac{40}{3} + 2 y - 2 = 0$

$x = \frac{y}{6} + \frac{8}{3}$

$\frac{5 y}{6} + \frac{40}{3} + 2 y - 2 = 0$

$x = \frac{y}{6} + \frac{8}{3}$

$\frac{5 y}{6} + \frac{40}{3} + 2 y - 2 = 0$

$x = \frac{y}{6} + \frac{8}{3}$

Step 2.2.1.2

To write $2 y$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$\frac{5 y}{6} + 2 y \cdot \frac{6}{6} + \frac{40}{3} - 2 = 0$

$x = \frac{y}{6} + \frac{8}{3}$

Step 2.2.1.3

Combine $2 y$ and $\frac{6}{6}$ .

$\frac{5 y}{6} + \frac{2 y \cdot 6}{6} + \frac{40}{3} - 2 = 0$

$x = \frac{y}{6} + \frac{8}{3}$

Step 2.2.1.4

Combine the numerators over the common denominator.

$\frac{5 y + 2 y \cdot 6}{6} + \frac{40}{3} - 2 = 0$

$x = \frac{y}{6} + \frac{8}{3}$

Step 2.2.1.5

Find the common denominator.

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Step 2.2.1.5.1

Multiply $\frac{40}{3}$ by $\frac{2}{2}$ .

$\frac{5 y + 2 y \cdot 6}{6} + \frac{40}{3} \cdot \frac{2}{2} - 2 = 0$

$x = \frac{y}{6} + \frac{8}{3}$

Step 2.2.1.5.2

Multiply $\frac{40}{3}$ by $\frac{2}{2}$ .

$\frac{5 y + 2 y \cdot 6}{6} + \frac{40 \cdot 2}{3 \cdot 2} - 2 = 0$

$x = \frac{y}{6} + \frac{8}{3}$

Step 2.2.1.5.3

Write $- 2$ as a fraction with denominator $1$ .

$\frac{5 y + 2 y \cdot 6}{6} + \frac{40 \cdot 2}{3 \cdot 2} + \frac{- 2}{1} = 0$

$x = \frac{y}{6} + \frac{8}{3}$

Step 2.2.1.5.4

Multiply $\frac{- 2}{1}$ by $\frac{6}{6}$ .

$\frac{5 y + 2 y \cdot 6}{6} + \frac{40 \cdot 2}{3 \cdot 2} + \frac{- 2}{1} \cdot \frac{6}{6} = 0$

$x = \frac{y}{6} + \frac{8}{3}$

Step 2.2.1.5.5

Multiply $\frac{- 2}{1}$ by $\frac{6}{6}$ .

$\frac{5 y + 2 y \cdot 6}{6} + \frac{40 \cdot 2}{3 \cdot 2} + \frac{- 2 \cdot 6}{6} = 0$

$x = \frac{y}{6} + \frac{8}{3}$

Step 2.2.1.5.6

Reorder the factors of $3 \cdot 2$ .

$\frac{5 y + 2 y \cdot 6}{6} + \frac{40 \cdot 2}{2 \cdot 3} + \frac{- 2 \cdot 6}{6} = 0$

$x = \frac{y}{6} + \frac{8}{3}$

Step 2.2.1.5.7

Multiply $2$ by $3$ .

$\frac{5 y + 2 y \cdot 6}{6} + \frac{40 \cdot 2}{6} + \frac{- 2 \cdot 6}{6} = 0$

$x = \frac{y}{6} + \frac{8}{3}$

$\frac{5 y + 2 y \cdot 6}{6} + \frac{40 \cdot 2}{6} + \frac{- 2 \cdot 6}{6} = 0$

$x = \frac{y}{6} + \frac{8}{3}$

Step 2.2.1.6

Combine the numerators over the common denominator.

$\frac{5 y + 2 y \cdot 6 + 40 \cdot 2 - 2 \cdot 6}{6} = 0$

$x = \frac{y}{6} + \frac{8}{3}$

Step 2.2.1.7

Simplify each term.

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Step 2.2.1.7.1

Multiply $6$ by $2$ .

$\frac{5 y + 12 y + 40 \cdot 2 - 2 \cdot 6}{6} = 0$

$x = \frac{y}{6} + \frac{8}{3}$

Step 2.2.1.7.2

Multiply $40$ by $2$ .

$\frac{5 y + 12 y + 80 - 2 \cdot 6}{6} = 0$

$x = \frac{y}{6} + \frac{8}{3}$

Step 2.2.1.7.3

Multiply $- 2$ by $6$ .

$\frac{5 y + 12 y + 80 - 12}{6} = 0$

$x = \frac{y}{6} + \frac{8}{3}$

$\frac{5 y + 12 y + 80 - 12}{6} = 0$

$x = \frac{y}{6} + \frac{8}{3}$

Step 2.2.1.8

Simplify terms.

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Step 2.2.1.8.1

Add $5 y$ and $12 y$ .

$\frac{17 y + 80 - 12}{6} = 0$

$x = \frac{y}{6} + \frac{8}{3}$

Step 2.2.1.8.2

Subtract $12$ from $80$ .

$\frac{17 y + 68}{6} = 0$

$x = \frac{y}{6} + \frac{8}{3}$

Step 2.2.1.8.3

Factor $17$ out of $17 y + 68$ .

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Step 2.2.1.8.3.1

Factor $17$ out of $17 y$ .

$\frac{17 (y) + 68}{6} = 0$

$x = \frac{y}{6} + \frac{8}{3}$

Step 2.2.1.8.3.2

Factor $17$ out of $68$ .

$\frac{17 y + 17 \cdot 4}{6} = 0$

$x = \frac{y}{6} + \frac{8}{3}$

Step 2.2.1.8.3.3

Factor $17$ out of $17 y + 17 \cdot 4$ .

$\frac{17 (y + 4)}{6} = 0$

$x = \frac{y}{6} + \frac{8}{3}$

$\frac{17 (y + 4)}{6} = 0$

$x = \frac{y}{6} + \frac{8}{3}$

$\frac{17 (y + 4)}{6} = 0$

$x = \frac{y}{6} + \frac{8}{3}$

$\frac{17 (y + 4)}{6} = 0$

$x = \frac{y}{6} + \frac{8}{3}$

$\frac{17 (y + 4)}{6} = 0$

$x = \frac{y}{6} + \frac{8}{3}$

$\frac{17 (y + 4)}{6} = 0$

$x = \frac{y}{6} + \frac{8}{3}$

Step 3

Solve for $y$ in $\frac{17 (y + 4)}{6} = 0$ .

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Step 3.1

Set the numerator equal to zero.

$17 (y + 4) = 0$

$x = \frac{y}{6} + \frac{8}{3}$

Step 3.2

Solve the equation for $y$ .

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Step 3.2.1

Divide each term in $17 (y + 4) = 0$ by $17$ and simplify.

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Step 3.2.1.1

Divide each term in $17 (y + 4) = 0$ by $17$ .

$\frac{17 (y + 4)}{17} = \frac{0}{17}$

$x = \frac{y}{6} + \frac{8}{3}$

Step 3.2.1.2

Simplify the left side.

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Step 3.2.1.2.1

Cancel the common factor of $17$ .

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Step 3.2.1.2.1.1

Cancel the common factor.

$\frac{17 (y + 4)}{17} = \frac{0}{17}$

$x = \frac{y}{6} + \frac{8}{3}$

Step 3.2.1.2.1.2

Divide $y + 4$ by $1$ .

$y + 4 = \frac{0}{17}$

$x = \frac{y}{6} + \frac{8}{3}$

$y + 4 = \frac{0}{17}$

$x = \frac{y}{6} + \frac{8}{3}$

$y + 4 = \frac{0}{17}$

$x = \frac{y}{6} + \frac{8}{3}$

Step 3.2.1.3

Simplify the right side.

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Step 3.2.1.3.1

Divide $0$ by $17$ .

$y + 4 = 0$

$x = \frac{y}{6} + \frac{8}{3}$

$y + 4 = 0$

$x = \frac{y}{6} + \frac{8}{3}$

$y + 4 = 0$

$x = \frac{y}{6} + \frac{8}{3}$

Step 3.2.2

Subtract $4$ from both sides of the equation.

$y = - 4$

$x = \frac{y}{6} + \frac{8}{3}$

$y = - 4$

$x = \frac{y}{6} + \frac{8}{3}$

$y = - 4$

$x = \frac{y}{6} + \frac{8}{3}$

Step 4

Replace all occurrences of $y$ with $- 4$ in each equation.

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Step 4.1

Replace all occurrences of $y$ in $x = \frac{y}{6} + \frac{8}{3}$ with $- 4$ .

$x = \frac{- 4}{6} + \frac{8}{3}$

$y = - 4$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $\frac{- 4}{6} + \frac{8}{3}$ .

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Step 4.2.1.1

Simplify each term.

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Step 4.2.1.1.1

Cancel the common factor of $- 4$ and $6$ .

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Step 4.2.1.1.1.1

Factor $2$ out of $- 4$ .

$x = \frac{2 (- 2)}{6} + \frac{8}{3}$

$y = - 4$

Step 4.2.1.1.1.2

Cancel the common factors.

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Step 4.2.1.1.1.2.1

Factor $2$ out of $6$ .

$x = \frac{2 \cdot - 2}{2 \cdot 3} + \frac{8}{3}$

$y = - 4$

Step 4.2.1.1.1.2.2

Cancel the common factor.

$x = \frac{2 \cdot - 2}{2 \cdot 3} + \frac{8}{3}$

$y = - 4$

Step 4.2.1.1.1.2.3

Rewrite the expression.

$x = \frac{- 2}{3} + \frac{8}{3}$

$y = - 4$

$x = \frac{- 2}{3} + \frac{8}{3}$

$y = - 4$

$x = \frac{- 2}{3} + \frac{8}{3}$

$y = - 4$

Step 4.2.1.1.2

Move the negative in front of the fraction.

$x = - \frac{2}{3} + \frac{8}{3}$

$y = - 4$

$x = - \frac{2}{3} + \frac{8}{3}$

$y = - 4$

Step 4.2.1.2

Combine fractions.

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Step 4.2.1.2.1

Combine the numerators over the common denominator.

$x = \frac{- 2 + 8}{3}$

$y = - 4$

Step 4.2.1.2.2

Simplify the expression.

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Step 4.2.1.2.2.1

Add $- 2$ and $8$ .

$x = \frac{6}{3}$

$y = - 4$

Step 4.2.1.2.2.2

Divide $6$ by $3$ .

$x = 2$

$y = - 4$

$x = 2$

$y = - 4$

$x = 2$

$y = - 4$

$x = 2$

$y = - 4$

$x = 2$

$y = - 4$

$x = 2$

$y = - 4$

Step 5

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

$(2, - 4)$

Step 6

The result can be shown in multiple forms.

Point Form:

$(2, - 4)$

Equation Form:

$x = 2, y = - 4$

Step 7