Finite Math Examples

$6 x - 4 y = 14$ , $5 x + 7 y = - 9$

Step 1

Solve for $x$ in $6 x - 4 y = 14$ .

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Step 1.1

Add $4 y$ to both sides of the equation.

$6 x = 14 + 4 y$

$5 x + 7 y = - 9$

Step 1.2

Divide each term in $6 x = 14 + 4 y$ by $6$ and simplify.

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Step 1.2.1

Divide each term in $6 x = 14 + 4 y$ by $6$ .

$\frac{6 x}{6} = \frac{14}{6} + \frac{4 y}{6}$

$5 x + 7 y = - 9$

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.

$\frac{6 x}{6} = \frac{14}{6} + \frac{4 y}{6}$

$5 x + 7 y = - 9$

Step 1.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{14}{6} + \frac{4 y}{6}$

$5 x + 7 y = - 9$

$x = \frac{14}{6} + \frac{4 y}{6}$

$5 x + 7 y = - 9$

$x = \frac{14}{6} + \frac{4 y}{6}$

$5 x + 7 y = - 9$

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 $14$ and $6$ .

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Step 1.2.3.1.1.1

Factor $2$ out of $14$ .

$x = \frac{2 (7)}{6} + \frac{4 y}{6}$

$5 x + 7 y = - 9$

Step 1.2.3.1.1.2

Cancel the common factors.

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Step 1.2.3.1.1.2.1

Factor $2$ out of $6$ .

$x = \frac{2 \cdot 7}{2 \cdot 3} + \frac{4 y}{6}$

$5 x + 7 y = - 9$

Step 1.2.3.1.1.2.2

Cancel the common factor.

$x = \frac{2 \cdot 7}{2 \cdot 3} + \frac{4 y}{6}$

$5 x + 7 y = - 9$

Step 1.2.3.1.1.2.3

Rewrite the expression.

$x = \frac{7}{3} + \frac{4 y}{6}$

$5 x + 7 y = - 9$

$x = \frac{7}{3} + \frac{4 y}{6}$

$5 x + 7 y = - 9$

$x = \frac{7}{3} + \frac{4 y}{6}$

$5 x + 7 y = - 9$

Step 1.2.3.1.2

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

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Step 1.2.3.1.2.1

Factor $2$ out of $4 y$ .

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

$5 x + 7 y = - 9$

Step 1.2.3.1.2.2

Cancel the common factors.

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Step 1.2.3.1.2.2.1

Factor $2$ out of $6$ .

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

$5 x + 7 y = - 9$

Step 1.2.3.1.2.2.2

Cancel the common factor.

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

$5 x + 7 y = - 9$

Step 1.2.3.1.2.2.3

Rewrite the expression.

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

$5 x + 7 y = - 9$

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

$5 x + 7 y = - 9$

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

$5 x + 7 y = - 9$

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

$5 x + 7 y = - 9$

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

$5 x + 7 y = - 9$

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

$5 x + 7 y = - 9$

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

$5 x + 7 y = - 9$

Step 2

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

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Step 2.1

Replace all occurrences of $x$ in $5 x + 7 y = - 9$ with $\frac{7}{3} + \frac{2 y}{3}$ .

$5 (\frac{7}{3} + \frac{2 y}{3}) + 7 y = - 9$

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

Step 2.2

Simplify the left side.

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Step 2.2.1

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

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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.

$5 (\frac{7}{3}) + 5 (\frac{2 y}{3}) + 7 y = - 9$

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

Step 2.2.1.1.2

Multiply $5 (\frac{7}{3})$ .

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Step 2.2.1.1.2.1

Combine $5$ and $\frac{7}{3}$ .

$\frac{5 \cdot 7}{3} + 5 (\frac{2 y}{3}) + 7 y = - 9$

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

Step 2.2.1.1.2.2

Multiply $5$ by $7$ .

$\frac{35}{3} + 5 (\frac{2 y}{3}) + 7 y = - 9$

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

$\frac{35}{3} + 5 (\frac{2 y}{3}) + 7 y = - 9$

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

Step 2.2.1.1.3

Multiply $5 \frac{2 y}{3}$ .

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Step 2.2.1.1.3.1

Combine $5$ and $\frac{2 y}{3}$ .

$\frac{35}{3} + \frac{5 (2 y)}{3} + 7 y = - 9$

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

Step 2.2.1.1.3.2

Multiply $2$ by $5$ .

$\frac{35}{3} + \frac{10 y}{3} + 7 y = - 9$

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

$\frac{35}{3} + \frac{10 y}{3} + 7 y = - 9$

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

$\frac{35}{3} + \frac{10 y}{3} + 7 y = - 9$

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

Step 2.2.1.2

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

$\frac{35}{3} + \frac{10 y}{3} + 7 y \cdot \frac{3}{3} = - 9$

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

Step 2.2.1.3

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

$\frac{35}{3} + \frac{10 y}{3} + \frac{7 y \cdot 3}{3} = - 9$

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

Step 2.2.1.4

Combine the numerators over the common denominator.

$\frac{35}{3} + \frac{10 y + 7 y \cdot 3}{3} = - 9$

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

Step 2.2.1.5

Combine the numerators over the common denominator.

$\frac{35 + 10 y + 7 y \cdot 3}{3} = - 9$

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

Step 2.2.1.6

Multiply $3$ by $7$ .

$\frac{35 + 10 y + 21 y}{3} = - 9$

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

Step 2.2.1.7

Add $10 y$ and $21 y$ .

$\frac{35 + 31 y}{3} = - 9$

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

$\frac{35 + 31 y}{3} = - 9$

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

$\frac{35 + 31 y}{3} = - 9$

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

$\frac{35 + 31 y}{3} = - 9$

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

Step 3

Solve for $y$ in $\frac{35 + 31 y}{3} = - 9$ .

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Step 3.1

Multiply both sides by $3$ .

$\frac{35 + 31 y}{3} \cdot 3 = - 9 \cdot 3$

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

Step 3.2

Simplify.

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Step 3.2.1

Simplify the left side.

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Step 3.2.1.1

Simplify $\frac{35 + 31 y}{3} \cdot 3$ .

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Step 3.2.1.1.1

Cancel the common factor of $3$ .

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Step 3.2.1.1.1.1

Cancel the common factor.

$\frac{35 + 31 y}{3} \cdot 3 = - 9 \cdot 3$

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

Step 3.2.1.1.1.2

Rewrite the expression.

$35 + 31 y = - 9 \cdot 3$

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

$35 + 31 y = - 9 \cdot 3$

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

Step 3.2.1.1.2

Reorder $35$ and $31 y$ .

$31 y + 35 = - 9 \cdot 3$

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

$31 y + 35 = - 9 \cdot 3$

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

$31 y + 35 = - 9 \cdot 3$

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

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Multiply $- 9$ by $3$ .

$31 y + 35 = - 27$

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

$31 y + 35 = - 27$

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

$31 y + 35 = - 27$

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

Step 3.3

Solve for $y$ .

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Step 3.3.1

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

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Step 3.3.1.1

Subtract $35$ from both sides of the equation.

$31 y = - 27 - 35$

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

Step 3.3.1.2

Subtract $35$ from $- 27$ .

$31 y = - 62$

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

$31 y = - 62$

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

Step 3.3.2

Divide each term in $31 y = - 62$ by $31$ and simplify.

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Step 3.3.2.1

Divide each term in $31 y = - 62$ by $31$ .

$\frac{31 y}{31} = \frac{- 62}{31}$

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

Step 3.3.2.2

Simplify the left side.

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Step 3.3.2.2.1

Cancel the common factor of $31$ .

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Step 3.3.2.2.1.1

Cancel the common factor.

$\frac{31 y}{31} = \frac{- 62}{31}$

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

Step 3.3.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 62}{31}$

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

$y = \frac{- 62}{31}$

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

$y = \frac{- 62}{31}$

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

Step 3.3.2.3

Simplify the right side.

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Step 3.3.2.3.1

Divide $- 62$ by $31$ .

$y = - 2$

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

$y = - 2$

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

$y = - 2$

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

$y = - 2$

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

$y = - 2$

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

Step 4

Replace all occurrences of $y$ with $- 2$ in each equation.

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Step 4.1

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

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

$y = - 2$

Step 4.2

Simplify the right side.

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Step 4.2.1

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

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Step 4.2.1.1

Combine the numerators over the common denominator.

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

$y = - 2$

Step 4.2.1.2

Simplify the expression.

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Step 4.2.1.2.1

Multiply $2$ by $- 2$ .

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

$y = - 2$

Step 4.2.1.2.2

Subtract $4$ from $7$ .

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

$y = - 2$

Step 4.2.1.2.3

Divide $3$ by $3$ .

$x = 1$

$y = - 2$

$x = 1$

$y = - 2$

$x = 1$

$y = - 2$

$x = 1$

$y = - 2$

$x = 1$

$y = - 2$

Step 5

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

$(1, - 2)$

Step 6

The result can be shown in multiple forms.

Point Form:

$(1, - 2)$

Equation Form:

$x = 1, y = - 2$

Step 7