Finite Math Examples

$6 x + 7 y = 9$ , $2 x - 7 y = - 25$

Step 1

Solve for $x$ in $6 x + 7 y = 9$ .

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Step 1.1

Subtract $7 y$ from both sides of the equation.

$6 x = 9 - 7 y$

$2 x - 7 y = - 25$

Step 1.2

Divide each term in $6 x = 9 - 7 y$ by $6$ and simplify.

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Step 1.2.1

Divide each term in $6 x = 9 - 7 y$ by $6$ .

$\frac{6 x}{6} = \frac{9}{6} + \frac{- 7 y}{6}$

$2 x - 7 y = - 25$

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.

$\frac{6 x}{6} = \frac{9}{6} + \frac{- 7 y}{6}$

$2 x - 7 y = - 25$

Step 1.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{9}{6} + \frac{- 7 y}{6}$

$2 x - 7 y = - 25$

$x = \frac{9}{6} + \frac{- 7 y}{6}$

$2 x - 7 y = - 25$

$x = \frac{9}{6} + \frac{- 7 y}{6}$

$2 x - 7 y = - 25$

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

Simplify each term.

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Step 1.2.3.1.1

Cancel the common factor of $9$ and $6$ .

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Step 1.2.3.1.1.1

Factor $3$ out of $9$ .

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

$2 x - 7 y = - 25$

Step 1.2.3.1.1.2

Cancel the common factors.

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Step 1.2.3.1.1.2.1

Factor $3$ out of $6$ .

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

$2 x - 7 y = - 25$

Step 1.2.3.1.1.2.2

Cancel the common factor.

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

$2 x - 7 y = - 25$

Step 1.2.3.1.1.2.3

Rewrite the expression.

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

$2 x - 7 y = - 25$

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

$2 x - 7 y = - 25$

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

$2 x - 7 y = - 25$

Step 1.2.3.1.2

Move the negative in front of the fraction.

$x = \frac{3}{2} - \frac{7 y}{6}$

$2 x - 7 y = - 25$

$x = \frac{3}{2} - \frac{7 y}{6}$

$2 x - 7 y = - 25$

$x = \frac{3}{2} - \frac{7 y}{6}$

$2 x - 7 y = - 25$

$x = \frac{3}{2} - \frac{7 y}{6}$

$2 x - 7 y = - 25$

$x = \frac{3}{2} - \frac{7 y}{6}$

$2 x - 7 y = - 25$

Step 2

Replace all occurrences of $x$ with $\frac{3}{2} - \frac{7 y}{6}$ in each equation.

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Step 2.1

Replace all occurrences of $x$ in $2 x - 7 y = - 25$ with $\frac{3}{2} - \frac{7 y}{6}$ .

$2 (\frac{3}{2} - \frac{7 y}{6}) - 7 y = - 25$

$x = \frac{3}{2} - \frac{7 y}{6}$

Step 2.2

Simplify the left side.

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Step 2.2.1

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

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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.

$2 (\frac{3}{2}) + 2 (- \frac{7 y}{6}) - 7 y = - 25$

$x = \frac{3}{2} - \frac{7 y}{6}$

Step 2.2.1.1.2

Cancel the common factor of $2$ .

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Step 2.2.1.1.2.1

Cancel the common factor.

$2 (\frac{3}{2}) + 2 (- \frac{7 y}{6}) - 7 y = - 25$

$x = \frac{3}{2} - \frac{7 y}{6}$

Step 2.2.1.1.2.2

Rewrite the expression.

$3 + 2 (- \frac{7 y}{6}) - 7 y = - 25$

$x = \frac{3}{2} - \frac{7 y}{6}$

$3 + 2 (- \frac{7 y}{6}) - 7 y = - 25$

$x = \frac{3}{2} - \frac{7 y}{6}$

Step 2.2.1.1.3

Cancel the common factor of $2$ .

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Step 2.2.1.1.3.1

Move the leading negative in $- \frac{7 y}{6}$ into the numerator.

$3 + 2 (\frac{- 7 y}{6}) - 7 y = - 25$

$x = \frac{3}{2} - \frac{7 y}{6}$

Step 2.2.1.1.3.2

Factor $2$ out of $6$ .

$3 + 2 (\frac{- 7 y}{2 (3)}) - 7 y = - 25$

$x = \frac{3}{2} - \frac{7 y}{6}$

Step 2.2.1.1.3.3

Cancel the common factor.

$3 + 2 (\frac{- 7 y}{2 \cdot 3}) - 7 y = - 25$

$x = \frac{3}{2} - \frac{7 y}{6}$

Step 2.2.1.1.3.4

Rewrite the expression.

$3 + \frac{- 7 y}{3} - 7 y = - 25$

$x = \frac{3}{2} - \frac{7 y}{6}$

$3 + \frac{- 7 y}{3} - 7 y = - 25$

$x = \frac{3}{2} - \frac{7 y}{6}$

Step 2.2.1.1.4

Move the negative in front of the fraction.

$3 - \frac{7 y}{3} - 7 y = - 25$

$x = \frac{3}{2} - \frac{7 y}{6}$

$3 - \frac{7 y}{3} - 7 y = - 25$

$x = \frac{3}{2} - \frac{7 y}{6}$

Step 2.2.1.2

To write $- 7 y$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$3 - \frac{7 y}{3} - 7 y \cdot \frac{3}{3} = - 25$

$x = \frac{3}{2} - \frac{7 y}{6}$

Step 2.2.1.3

Simplify terms.

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Step 2.2.1.3.1

Combine $- 7 y$ and $\frac{3}{3}$ .

$3 - \frac{7 y}{3} + \frac{- 7 y \cdot 3}{3} = - 25$

$x = \frac{3}{2} - \frac{7 y}{6}$

Step 2.2.1.3.2

Combine the numerators over the common denominator.

$3 + \frac{- 7 y - 7 y \cdot 3}{3} = - 25$

$x = \frac{3}{2} - \frac{7 y}{6}$

$3 + \frac{- 7 y - 7 y \cdot 3}{3} = - 25$

$x = \frac{3}{2} - \frac{7 y}{6}$

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 $7 y$ out of $- 7 y - 7 y \cdot 3$ .

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Step 2.2.1.4.1.1.1

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

$3 + \frac{7 y (- 1) - 7 y \cdot 3}{3} = - 25$

$x = \frac{3}{2} - \frac{7 y}{6}$

Step 2.2.1.4.1.1.2

Factor $7 y$ out of $- 7 y \cdot 3$ .

$3 + \frac{7 y (- 1) + 7 y (- 1 \cdot 3)}{3} = - 25$

$x = \frac{3}{2} - \frac{7 y}{6}$

Step 2.2.1.4.1.1.3

Factor $7 y$ out of $7 y (- 1) + 7 y (- 1 \cdot 3)$ .

$3 + \frac{7 y (- 1 - 1 \cdot 3)}{3} = - 25$

$x = \frac{3}{2} - \frac{7 y}{6}$

$3 + \frac{7 y (- 1 - 1 \cdot 3)}{3} = - 25$

$x = \frac{3}{2} - \frac{7 y}{6}$

Step 2.2.1.4.1.2

Multiply $- 1$ by $3$ .

$3 + \frac{7 y (- 1 - 3)}{3} = - 25$

$x = \frac{3}{2} - \frac{7 y}{6}$

Step 2.2.1.4.1.3

Subtract $3$ from $- 1$ .

$3 + \frac{7 y \cdot - 4}{3} = - 25$

$x = \frac{3}{2} - \frac{7 y}{6}$

Step 2.2.1.4.1.4

Multiply $- 4$ by $7$ .

$3 + \frac{- 28 y}{3} = - 25$

$x = \frac{3}{2} - \frac{7 y}{6}$

$3 + \frac{- 28 y}{3} = - 25$

$x = \frac{3}{2} - \frac{7 y}{6}$

Step 2.2.1.4.2

Move the negative in front of the fraction.

$3 - \frac{28 y}{3} = - 25$

$x = \frac{3}{2} - \frac{7 y}{6}$

$3 - \frac{28 y}{3} = - 25$

$x = \frac{3}{2} - \frac{7 y}{6}$

$3 - \frac{28 y}{3} = - 25$

$x = \frac{3}{2} - \frac{7 y}{6}$

$3 - \frac{28 y}{3} = - 25$

$x = \frac{3}{2} - \frac{7 y}{6}$

$3 - \frac{28 y}{3} = - 25$

$x = \frac{3}{2} - \frac{7 y}{6}$

Step 3

Solve for $y$ in $3 - \frac{28 y}{3} = - 25$ .

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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 $3$ from both sides of the equation.

$- \frac{28 y}{3} = - 25 - 3$

$x = \frac{3}{2} - \frac{7 y}{6}$

Step 3.1.2

Subtract $3$ from $- 25$ .

$- \frac{28 y}{3} = - 28$

$x = \frac{3}{2} - \frac{7 y}{6}$

$- \frac{28 y}{3} = - 28$

$x = \frac{3}{2} - \frac{7 y}{6}$

Step 3.2

Multiply both sides of the equation by $- \frac{3}{28}$ .

$- \frac{3}{28} \cdot (- \frac{28 y}{3}) = - \frac{3}{28} \cdot - 28$

$x = \frac{3}{2} - \frac{7 y}{6}$

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 $- \frac{3}{28} (- \frac{28 y}{3})$ .

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Step 3.3.1.1.1

Cancel the common factor of $3$ .

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Step 3.3.1.1.1.1

Move the leading negative in $- \frac{3}{28}$ into the numerator.

$\frac{- 3}{28} \cdot (- \frac{28 y}{3}) = - \frac{3}{28} \cdot - 28$

$x = \frac{3}{2} - \frac{7 y}{6}$

Step 3.3.1.1.1.2

Move the leading negative in $- \frac{28 y}{3}$ into the numerator.

$\frac{- 3}{28} \cdot \frac{- 28 y}{3} = - \frac{3}{28} \cdot - 28$

$x = \frac{3}{2} - \frac{7 y}{6}$

Step 3.3.1.1.1.3

Factor $3$ out of $- 3$ .

$\frac{3 (- 1)}{28} \cdot \frac{- 28 y}{3} = - \frac{3}{28} \cdot - 28$

$x = \frac{3}{2} - \frac{7 y}{6}$

Step 3.3.1.1.1.4

Cancel the common factor.

$\frac{3 \cdot - 1}{28} \cdot \frac{- 28 y}{3} = - \frac{3}{28} \cdot - 28$

$x = \frac{3}{2} - \frac{7 y}{6}$

Step 3.3.1.1.1.5

Rewrite the expression.

$\frac{- 1}{28} \cdot (- 28 y) = - \frac{3}{28} \cdot - 28$

$x = \frac{3}{2} - \frac{7 y}{6}$

$\frac{- 1}{28} \cdot (- 28 y) = - \frac{3}{28} \cdot - 28$

$x = \frac{3}{2} - \frac{7 y}{6}$

Step 3.3.1.1.2

Cancel the common factor of $28$ .

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Step 3.3.1.1.2.1

Factor $28$ out of $- 28 y$ .

$\frac{- 1}{28} \cdot (28 (- y)) = - \frac{3}{28} \cdot - 28$

$x = \frac{3}{2} - \frac{7 y}{6}$

Step 3.3.1.1.2.2

Cancel the common factor.

$\frac{- 1}{28} \cdot (28 (- y)) = - \frac{3}{28} \cdot - 28$

$x = \frac{3}{2} - \frac{7 y}{6}$

Step 3.3.1.1.2.3

Rewrite the expression.

$y = - \frac{3}{28} \cdot - 28$

$x = \frac{3}{2} - \frac{7 y}{6}$

$y = - \frac{3}{28} \cdot - 28$

$x = \frac{3}{2} - \frac{7 y}{6}$

Step 3.3.1.1.3

Multiply.

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Step 3.3.1.1.3.1

Multiply $- 1$ by $- 1$ .

$1 y = - \frac{3}{28} \cdot - 28$

$x = \frac{3}{2} - \frac{7 y}{6}$

Step 3.3.1.1.3.2

Multiply $y$ by $1$ .

$y = - \frac{3}{28} \cdot - 28$

$x = \frac{3}{2} - \frac{7 y}{6}$

$y = - \frac{3}{28} \cdot - 28$

$x = \frac{3}{2} - \frac{7 y}{6}$

$y = - \frac{3}{28} \cdot - 28$

$x = \frac{3}{2} - \frac{7 y}{6}$

$y = - \frac{3}{28} \cdot - 28$

$x = \frac{3}{2} - \frac{7 y}{6}$

Step 3.3.2

Simplify the right side.

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Step 3.3.2.1

Simplify $- \frac{3}{28} \cdot - 28$ .

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Step 3.3.2.1.1

Cancel the common factor of $28$ .

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Step 3.3.2.1.1.1

Move the leading negative in $- \frac{3}{28}$ into the numerator.

$y = \frac{- 3}{28} \cdot - 28$

$x = \frac{3}{2} - \frac{7 y}{6}$

Step 3.3.2.1.1.2

Factor $28$ out of $- 28$ .

$y = \frac{- 3}{28} \cdot (28 (- 1))$

$x = \frac{3}{2} - \frac{7 y}{6}$

Step 3.3.2.1.1.3

Cancel the common factor.

$y = \frac{- 3}{28} \cdot (28 \cdot - 1)$

$x = \frac{3}{2} - \frac{7 y}{6}$

Step 3.3.2.1.1.4

Rewrite the expression.

$y = - 3 \cdot - 1$

$x = \frac{3}{2} - \frac{7 y}{6}$

$y = - 3 \cdot - 1$

$x = \frac{3}{2} - \frac{7 y}{6}$

Step 3.3.2.1.2

Multiply $- 3$ by $- 1$ .

$y = 3$

$x = \frac{3}{2} - \frac{7 y}{6}$

$y = 3$

$x = \frac{3}{2} - \frac{7 y}{6}$

$y = 3$

$x = \frac{3}{2} - \frac{7 y}{6}$

$y = 3$

$x = \frac{3}{2} - \frac{7 y}{6}$

$y = 3$

$x = \frac{3}{2} - \frac{7 y}{6}$

Step 4

Replace all occurrences of $y$ with $3$ in each equation.

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Step 4.1

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

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

$y = 3$

Step 4.2

Simplify the right side.

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Step 4.2.1

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

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Step 4.2.1.1

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

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Step 4.2.1.1.1

Factor $3$ out of $7 (3)$ .

$x = \frac{3}{2} - \frac{3 \cdot 7}{6}$

$y = 3$

Step 4.2.1.1.2

Cancel the common factors.

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Step 4.2.1.1.2.1

Factor $3$ out of $6$ .

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

$y = 3$

Step 4.2.1.1.2.2

Cancel the common factor.

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

$y = 3$

Step 4.2.1.1.2.3

Rewrite the expression.

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

$y = 3$

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

$y = 3$

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

$y = 3$

Step 4.2.1.2

Combine the numerators over the common denominator.

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

$y = 3$

Step 4.2.1.3

Simplify the expression.

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Step 4.2.1.3.1

Subtract $7$ from $3$ .

$x = \frac{- 4}{2}$

$y = 3$

Step 4.2.1.3.2

Divide $- 4$ by $2$ .

$x = - 2$

$y = 3$

$x = - 2$

$y = 3$

$x = - 2$

$y = 3$

$x = - 2$

$y = 3$

$x = - 2$

$y = 3$

Step 5

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

$(- 2, 3)$

Step 6

The result can be shown in multiple forms.

Point Form:

$(- 2, 3)$

Equation Form:

$x = - 2, y = 3$

Step 7