Algebra Examples

$- 6 x + 7 y = 9$ and $2 x - 2 y = - 4$

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.

$- 6 x = 9 - 7 y$

$2 x - 2 y = - 4$

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 $- 6$ .

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

$2 x - 2 y = - 4$

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.

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

$2 x - 2 y = - 4$

Step 1.2.2.1.2

Divide $x$ by $1$ .

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

$2 x - 2 y = - 4$

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

$2 x - 2 y = - 4$

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

$2 x - 2 y = - 4$

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 $9$ .

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

$2 x - 2 y = - 4$

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 $- 6$ .

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

$2 x - 2 y = - 4$

Step 1.2.3.1.1.2.2

Cancel the common factor.

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

$2 x - 2 y = - 4$

Step 1.2.3.1.1.2.3

Rewrite the expression.

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

$2 x - 2 y = - 4$

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

$2 x - 2 y = - 4$

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

$2 x - 2 y = - 4$

Step 1.2.3.1.2

Move the negative in front of the fraction.

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

$2 x - 2 y = - 4$

Step 1.2.3.1.3

Dividing two negative values results in a positive value.

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

$2 x - 2 y = - 4$

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

$2 x - 2 y = - 4$

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

$2 x - 2 y = - 4$

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

$2 x - 2 y = - 4$

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

$2 x - 2 y = - 4$

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 $2 x - 2 y = - 4$ with $- \frac{3}{2} + \frac{7 y}{6}$ .

$2 (- \frac{3}{2} + \frac{7 y}{6}) - 2 y = - 4$

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

Step 2.2

Simplify the left side.

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Step 2.2.1

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

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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.

$2 (- \frac{3}{2}) + 2 (\frac{7 y}{6}) - 2 y = - 4$

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

Step 2.2.1.1.2

Cancel the common factor of $2$ .

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Step 2.2.1.1.2.1

Move the leading negative in $- \frac{3}{2}$ into the numerator.

$2 (\frac{- 3}{2}) + 2 (\frac{7 y}{6}) - 2 y = - 4$

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

Step 2.2.1.1.2.2

Cancel the common factor.

$2 (\frac{- 3}{2}) + 2 (\frac{7 y}{6}) - 2 y = - 4$

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

Step 2.2.1.1.2.3

Rewrite the expression.

$- 3 + 2 (\frac{7 y}{6}) - 2 y = - 4$

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

$- 3 + 2 (\frac{7 y}{6}) - 2 y = - 4$

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

Step 2.2.1.1.3

Cancel the common factor of $2$ .

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Step 2.2.1.1.3.1

Factor $2$ out of $6$ .

$- 3 + 2 (\frac{7 y}{2 (3)}) - 2 y = - 4$

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

Step 2.2.1.1.3.2

Cancel the common factor.

$- 3 + 2 (\frac{7 y}{2 \cdot 3}) - 2 y = - 4$

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

Step 2.2.1.1.3.3

Rewrite the expression.

$- 3 + \frac{7 y}{3} - 2 y = - 4$

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

$- 3 + \frac{7 y}{3} - 2 y = - 4$

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

$- 3 + \frac{7 y}{3} - 2 y = - 4$

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

Step 2.2.1.2

To write $- 2 y$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

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

Step 2.2.1.3

Simplify terms.

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Step 2.2.1.3.1

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

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

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

Step 2.2.1.3.2

Combine the numerators over the common denominator.

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

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

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

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

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 $y$ out of $7 y - 2 y \cdot 3$ .

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Step 2.2.1.4.1.1.1

Factor $y$ out of $7 y$ .

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

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

Step 2.2.1.4.1.1.2

Factor $y$ out of $- 2 y \cdot 3$ .

$- 3 + \frac{y \cdot 7 + y (- 2 \cdot 3)}{3} = - 4$

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

Step 2.2.1.4.1.1.3

Factor $y$ out of $y \cdot 7 + y (- 2 \cdot 3)$ .

$- 3 + \frac{y (7 - 2 \cdot 3)}{3} = - 4$

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

$- 3 + \frac{y (7 - 2 \cdot 3)}{3} = - 4$

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

Step 2.2.1.4.1.2

Multiply $- 2$ by $3$ .

$- 3 + \frac{y (7 - 6)}{3} = - 4$

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

Step 2.2.1.4.1.3

Subtract $6$ from $7$ .

$- 3 + \frac{y \cdot 1}{3} = - 4$

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

$- 3 + \frac{y \cdot 1}{3} = - 4$

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

Step 2.2.1.4.2

Multiply $y$ by $1$ .

$- 3 + \frac{y}{3} = - 4$

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

$- 3 + \frac{y}{3} = - 4$

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

$- 3 + \frac{y}{3} = - 4$

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

$- 3 + \frac{y}{3} = - 4$

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

$- 3 + \frac{y}{3} = - 4$

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

Step 3

Solve for $y$ in $- 3 + \frac{y}{3} = - 4$ .

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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

Add $3$ to both sides of the equation.

$\frac{y}{3} = - 4 + 3$

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

Step 3.1.2

Add $- 4$ and $3$ .

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

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

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

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

Step 3.2

Multiply both sides of the equation by $3$ .

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

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

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

Cancel the common factor of $3$ .

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Step 3.3.1.1.1

Cancel the common factor.

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

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

Step 3.3.1.1.2

Rewrite the expression.

$y = 3 \cdot - 1$

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

$y = 3 \cdot - 1$

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

$y = 3 \cdot - 1$

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

Step 3.3.2

Simplify the right side.

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Step 3.3.2.1

Multiply $3$ by $- 1$ .

$y = - 3$

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

$y = - 3$

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

$y = - 3$

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

$y = - 3$

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

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

Multiply $7$ by $- 3$ .

$x = - \frac{3}{2} + \frac{- 21}{6}$

$y = - 3$

Step 4.2.1.2

Simplify each term.

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Step 4.2.1.2.1

Cancel the common factor of $- 21$ and $6$ .

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Step 4.2.1.2.1.1

Factor $3$ out of $- 21$ .

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

$y = - 3$

Step 4.2.1.2.1.2

Cancel the common factors.

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Step 4.2.1.2.1.2.1

Factor $3$ out of $6$ .

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

$y = - 3$

Step 4.2.1.2.1.2.2

Cancel the common factor.

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

$y = - 3$

Step 4.2.1.2.1.2.3

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 4.2.1.2.2

Move the negative in front of the fraction.

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

$y = - 3$

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

$y = - 3$

Step 4.2.1.3

Combine fractions.

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Step 4.2.1.3.1

Combine the numerators over the common denominator.

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

$y = - 3$

Step 4.2.1.3.2

Simplify the expression.

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Step 4.2.1.3.2.1

Subtract $7$ from $- 3$ .

$x = \frac{- 10}{2}$

$y = - 3$

Step 4.2.1.3.2.2

Divide $- 10$ by $2$ .

$x = - 5$

$y = - 3$

$x = - 5$

$y = - 3$

$x = - 5$

$y = - 3$

$x = - 5$

$y = - 3$

$x = - 5$

$y = - 3$

$x = - 5$

$y = - 3$

Step 5

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

$(- 5, - 3)$

Step 6

The result can be shown in multiple forms.

Point Form:

$(- 5, - 3)$

Equation Form:

$x = - 5, y = - 3$

Step 7