Finite Math Examples

$2 x - 6 y = 7$ , $5 x + 2 y = 10$

Step 1

Solve for $x$ in $2 x - 6 y = 7$ .

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Step 1.1

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

$2 x = 7 + 6 y$

$5 x + 2 y = 10$

Step 1.2

Divide each term in $2 x = 7 + 6 y$ by $2$ and simplify.

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Step 1.2.1

Divide each term in $2 x = 7 + 6 y$ by $2$ .

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

$5 x + 2 y = 10$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $2$ .

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Step 1.2.2.1.1

Cancel the common factor.

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

$5 x + 2 y = 10$

Step 1.2.2.1.2

Divide $x$ by $1$ .

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

$5 x + 2 y = 10$

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

$5 x + 2 y = 10$

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

$5 x + 2 y = 10$

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

Cancel the common factor of $6$ and $2$ .

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Step 1.2.3.1.1

Factor $2$ out of $6 y$ .

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

$5 x + 2 y = 10$

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 $2$ .

$x = \frac{7}{2} + \frac{2 (3 y)}{2 (1)}$

$5 x + 2 y = 10$

Step 1.2.3.1.2.2

Cancel the common factor.

$x = \frac{7}{2} + \frac{2 (3 y)}{2 \cdot 1}$

$5 x + 2 y = 10$

Step 1.2.3.1.2.3

Rewrite the expression.

$x = \frac{7}{2} + \frac{3 y}{1}$

$5 x + 2 y = 10$

Step 1.2.3.1.2.4

Divide $3 y$ by $1$ .

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

$5 x + 2 y = 10$

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

$5 x + 2 y = 10$

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

$5 x + 2 y = 10$

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

$5 x + 2 y = 10$

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

$5 x + 2 y = 10$

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

$5 x + 2 y = 10$

Step 2

Replace all occurrences of $x$ with $\frac{7}{2} + 3 y$ in each equation.

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Step 2.1

Replace all occurrences of $x$ in $5 x + 2 y = 10$ with $\frac{7}{2} + 3 y$ .

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

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

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $5 (\frac{7}{2} + 3 y) + 2 y$ .

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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.

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

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

Step 2.2.1.1.2

Multiply $5 (\frac{7}{2})$ .

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Step 2.2.1.1.2.1

Combine $5$ and $\frac{7}{2}$ .

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

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

Step 2.2.1.1.2.2

Multiply $5$ by $7$ .

$\frac{35}{2} + 5 (3 y) + 2 y = 10$

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

$\frac{35}{2} + 5 (3 y) + 2 y = 10$

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

Step 2.2.1.1.3

Multiply $3$ by $5$ .

$\frac{35}{2} + 15 y + 2 y = 10$

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

$\frac{35}{2} + 15 y + 2 y = 10$

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

Step 2.2.1.2

Add $15 y$ and $2 y$ .

$\frac{35}{2} + 17 y = 10$

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

$\frac{35}{2} + 17 y = 10$

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

$\frac{35}{2} + 17 y = 10$

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

$\frac{35}{2} + 17 y = 10$

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

Step 3

Solve for $y$ in $\frac{35}{2} + 17 y = 10$ .

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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 $\frac{35}{2}$ from both sides of the equation.

$17 y = 10 - \frac{35}{2}$

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

Step 3.1.2

To write $10$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$17 y = 10 \cdot \frac{2}{2} - \frac{35}{2}$

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

Step 3.1.3

Combine $10$ and $\frac{2}{2}$ .

$17 y = \frac{10 \cdot 2}{2} - \frac{35}{2}$

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

Step 3.1.4

Combine the numerators over the common denominator.

$17 y = \frac{10 \cdot 2 - 35}{2}$

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

Step 3.1.5

Simplify the numerator.

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Step 3.1.5.1

Multiply $10$ by $2$ .

$17 y = \frac{20 - 35}{2}$

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

Step 3.1.5.2

Subtract $35$ from $20$ .

$17 y = \frac{- 15}{2}$

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

$17 y = \frac{- 15}{2}$

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

Step 3.1.6

Move the negative in front of the fraction.

$17 y = - \frac{15}{2}$

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

$17 y = - \frac{15}{2}$

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

Step 3.2

Divide each term in $17 y = - \frac{15}{2}$ by $17$ and simplify.

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Step 3.2.1

Divide each term in $17 y = - \frac{15}{2}$ by $17$ .

$\frac{17 y}{17} = \frac{- \frac{15}{2}}{17}$

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

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $17$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{17 y}{17} = \frac{- \frac{15}{2}}{17}$

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

Step 3.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{- \frac{15}{2}}{17}$

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

$y = \frac{- \frac{15}{2}}{17}$

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

$y = \frac{- \frac{15}{2}}{17}$

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

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.

$y = - \frac{15}{2} \cdot \frac{1}{17}$

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

Step 3.2.3.2

Multiply $- \frac{15}{2} \cdot \frac{1}{17}$ .

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Step 3.2.3.2.1

Multiply $\frac{1}{17}$ by $\frac{15}{2}$ .

$y = - \frac{15}{17 \cdot 2}$

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

Step 3.2.3.2.2

Multiply $17$ by $2$ .

$y = - \frac{15}{34}$

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

$y = - \frac{15}{34}$

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

$y = - \frac{15}{34}$

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

$y = - \frac{15}{34}$

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

$y = - \frac{15}{34}$

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

Step 4

Replace all occurrences of $y$ with $- \frac{15}{34}$ in each equation.

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Step 4.1

Replace all occurrences of $y$ in $x = \frac{7}{2} + 3 y$ with $- \frac{15}{34}$ .

$x = \frac{7}{2} + 3 (- \frac{15}{34})$

$y = - \frac{15}{34}$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $\frac{7}{2} + 3 (- \frac{15}{34})$ .

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Step 4.2.1.1

Simplify each term.

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Step 4.2.1.1.1

Multiply $3 (- \frac{15}{34})$ .

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Step 4.2.1.1.1.1

Multiply $- 1$ by $3$ .

$x = \frac{7}{2} - 3 (\frac{15}{34})$

$y = - \frac{15}{34}$

Step 4.2.1.1.1.2

Combine $- 3$ and $\frac{15}{34}$ .

$x = \frac{7}{2} + \frac{- 3 \cdot 15}{34}$

$y = - \frac{15}{34}$

Step 4.2.1.1.1.3

Multiply $- 3$ by $15$ .

$x = \frac{7}{2} + \frac{- 45}{34}$

$y = - \frac{15}{34}$

$x = \frac{7}{2} + \frac{- 45}{34}$

$y = - \frac{15}{34}$

Step 4.2.1.1.2

Move the negative in front of the fraction.

$x = \frac{7}{2} - \frac{45}{34}$

$y = - \frac{15}{34}$

$x = \frac{7}{2} - \frac{45}{34}$

$y = - \frac{15}{34}$

Step 4.2.1.2

To write $\frac{7}{2}$ as a fraction with a common denominator, multiply by $\frac{17}{17}$ .

$x = \frac{7}{2} \cdot \frac{17}{17} - \frac{45}{34}$

$y = - \frac{15}{34}$

Step 4.2.1.3

Write each expression with a common denominator of $34$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.2.1.3.1

Multiply $\frac{7}{2}$ by $\frac{17}{17}$ .

$x = \frac{7 \cdot 17}{2 \cdot 17} - \frac{45}{34}$

$y = - \frac{15}{34}$

Step 4.2.1.3.2

Multiply $2$ by $17$ .

$x = \frac{7 \cdot 17}{34} - \frac{45}{34}$

$y = - \frac{15}{34}$

$x = \frac{7 \cdot 17}{34} - \frac{45}{34}$

$y = - \frac{15}{34}$

Step 4.2.1.4

Combine the numerators over the common denominator.

$x = \frac{7 \cdot 17 - 45}{34}$

$y = - \frac{15}{34}$

Step 4.2.1.5

Simplify the numerator.

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Step 4.2.1.5.1

Multiply $7$ by $17$ .

$x = \frac{119 - 45}{34}$

$y = - \frac{15}{34}$

Step 4.2.1.5.2

Subtract $45$ from $119$ .

$x = \frac{74}{34}$

$y = - \frac{15}{34}$

$x = \frac{74}{34}$

$y = - \frac{15}{34}$

Step 4.2.1.6

Cancel the common factor of $74$ and $34$ .

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Step 4.2.1.6.1

Factor $2$ out of $74$ .

$x = \frac{2 (37)}{34}$

$y = - \frac{15}{34}$

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 $34$ .

$x = \frac{2 \cdot 37}{2 \cdot 17}$

$y = - \frac{15}{34}$

Step 4.2.1.6.2.2

Cancel the common factor.

$x = \frac{2 \cdot 37}{2 \cdot 17}$

$y = - \frac{15}{34}$

Step 4.2.1.6.2.3

Rewrite the expression.

$x = \frac{37}{17}$

$y = - \frac{15}{34}$

$x = \frac{37}{17}$

$y = - \frac{15}{34}$

$x = \frac{37}{17}$

$y = - \frac{15}{34}$

$x = \frac{37}{17}$

$y = - \frac{15}{34}$

$x = \frac{37}{17}$

$y = - \frac{15}{34}$

$x = \frac{37}{17}$

$y = - \frac{15}{34}$

Step 5

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

$(\frac{37}{17}, - \frac{15}{34})$

Step 6

The result can be shown in multiple forms.

Point Form:

$(\frac{37}{17}, - \frac{15}{34})$

Equation Form:

$x = \frac{37}{17}, y = - \frac{15}{34}$

Step 7