Algebra Examples

$5 x = - 3 y - 31$ $2 y = - 4 x - 22$

Step 1

Divide each term in $5 x = - 3 y - 31$ by $5$ and simplify.

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Step 1.1

Divide each term in $5 x = - 3 y - 31$ by $5$ .

$\frac{5 x}{5} = \frac{- 3 y}{5} + \frac{- 31}{5}$

$2 y = - 4 x - 22$

Step 1.2

Simplify the left side.

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Step 1.2.1

Cancel the common factor of $5$ .

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Step 1.2.1.1

Cancel the common factor.

$\frac{5 x}{5} = \frac{- 3 y}{5} + \frac{- 31}{5}$

$2 y = - 4 x - 22$

Step 1.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 3 y}{5} + \frac{- 31}{5}$

$2 y = - 4 x - 22$

$x = \frac{- 3 y}{5} + \frac{- 31}{5}$

$2 y = - 4 x - 22$

$x = \frac{- 3 y}{5} + \frac{- 31}{5}$

$2 y = - 4 x - 22$

Step 1.3

Simplify the right side.

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Step 1.3.1

Simplify each term.

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Step 1.3.1.1

Move the negative in front of the fraction.

$x = - \frac{3 y}{5} + \frac{- 31}{5}$

$2 y = - 4 x - 22$

Step 1.3.1.2

Move the negative in front of the fraction.

$x = - \frac{3 y}{5} - \frac{31}{5}$

$2 y = - 4 x - 22$

$x = - \frac{3 y}{5} - \frac{31}{5}$

$2 y = - 4 x - 22$

$x = - \frac{3 y}{5} - \frac{31}{5}$

$2 y = - 4 x - 22$

$x = - \frac{3 y}{5} - \frac{31}{5}$

$2 y = - 4 x - 22$

Step 2

Replace all occurrences of $x$ with $- \frac{3 y}{5} - \frac{31}{5}$ in each equation.

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Step 2.1

Replace all occurrences of $x$ in $2 y = - 4 x - 22$ with $- \frac{3 y}{5} - \frac{31}{5}$ .

$2 y = - 4 (- \frac{3 y}{5} - \frac{31}{5}) - 22$

$x = - \frac{3 y}{5} - \frac{31}{5}$

Step 2.2

Simplify the right side.

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Step 2.2.1

Simplify $- 4 (- \frac{3 y}{5} - \frac{31}{5}) - 22$ .

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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.

$2 y = - 4 (- \frac{3 y}{5}) - 4 (- \frac{31}{5}) - 22$

$x = - \frac{3 y}{5} - \frac{31}{5}$

Step 2.2.1.1.2

Multiply $- 4 (- \frac{3 y}{5})$ .

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Step 2.2.1.1.2.1

Multiply $- 1$ by $- 4$ .

$2 y = 4 (\frac{3 y}{5}) - 4 (- \frac{31}{5}) - 22$

$x = - \frac{3 y}{5} - \frac{31}{5}$

Step 2.2.1.1.2.2

Combine $4$ and $\frac{3 y}{5}$ .

$2 y = \frac{4 (3 y)}{5} - 4 (- \frac{31}{5}) - 22$

$x = - \frac{3 y}{5} - \frac{31}{5}$

Step 2.2.1.1.2.3

Multiply $3$ by $4$ .

$2 y = \frac{12 y}{5} - 4 (- \frac{31}{5}) - 22$

$x = - \frac{3 y}{5} - \frac{31}{5}$

$2 y = \frac{12 y}{5} - 4 (- \frac{31}{5}) - 22$

$x = - \frac{3 y}{5} - \frac{31}{5}$

Step 2.2.1.1.3

Multiply $- 4 (- \frac{31}{5})$ .

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Step 2.2.1.1.3.1

Multiply $- 1$ by $- 4$ .

$2 y = \frac{12 y}{5} + 4 (\frac{31}{5}) - 22$

$x = - \frac{3 y}{5} - \frac{31}{5}$

Step 2.2.1.1.3.2

Combine $4$ and $\frac{31}{5}$ .

$2 y = \frac{12 y}{5} + \frac{4 \cdot 31}{5} - 22$

$x = - \frac{3 y}{5} - \frac{31}{5}$

Step 2.2.1.1.3.3

Multiply $4$ by $31$ .

$2 y = \frac{12 y}{5} + \frac{124}{5} - 22$

$x = - \frac{3 y}{5} - \frac{31}{5}$

$2 y = \frac{12 y}{5} + \frac{124}{5} - 22$

$x = - \frac{3 y}{5} - \frac{31}{5}$

$2 y = \frac{12 y}{5} + \frac{124}{5} - 22$

$x = - \frac{3 y}{5} - \frac{31}{5}$

Step 2.2.1.2

To write $- 22$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$2 y = \frac{12 y}{5} + \frac{124}{5} - 22 \cdot \frac{5}{5}$

$x = - \frac{3 y}{5} - \frac{31}{5}$

Step 2.2.1.3

Combine $- 22$ and $\frac{5}{5}$ .

$2 y = \frac{12 y}{5} + \frac{124}{5} + \frac{- 22 \cdot 5}{5}$

$x = - \frac{3 y}{5} - \frac{31}{5}$

Step 2.2.1.4

Combine the numerators over the common denominator.

$2 y = \frac{12 y}{5} + \frac{124 - 22 \cdot 5}{5}$

$x = - \frac{3 y}{5} - \frac{31}{5}$

Step 2.2.1.5

Simplify the numerator.

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Step 2.2.1.5.1

Multiply $- 22$ by $5$ .

$2 y = \frac{12 y}{5} + \frac{124 - 110}{5}$

$x = - \frac{3 y}{5} - \frac{31}{5}$

Step 2.2.1.5.2

Subtract $110$ from $124$ .

$2 y = \frac{12 y}{5} + \frac{14}{5}$

$x = - \frac{3 y}{5} - \frac{31}{5}$

$2 y = \frac{12 y}{5} + \frac{14}{5}$

$x = - \frac{3 y}{5} - \frac{31}{5}$

$2 y = \frac{12 y}{5} + \frac{14}{5}$

$x = - \frac{3 y}{5} - \frac{31}{5}$

$2 y = \frac{12 y}{5} + \frac{14}{5}$

$x = - \frac{3 y}{5} - \frac{31}{5}$

$2 y = \frac{12 y}{5} + \frac{14}{5}$

$x = - \frac{3 y}{5} - \frac{31}{5}$

Step 3

Solve for $y$ in $2 y = \frac{12 y}{5} + \frac{14}{5}$ .

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Step 3.1

Move all terms containing $y$ to the left side of the equation.

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Step 3.1.1

Subtract $\frac{12 y}{5}$ from both sides of the equation.

$2 y - \frac{12 y}{5} = \frac{14}{5}$

$x = - \frac{3 y}{5} - \frac{31}{5}$

Step 3.1.2

To write $2 y$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$2 y \cdot \frac{5}{5} - \frac{12 y}{5} = \frac{14}{5}$

$x = - \frac{3 y}{5} - \frac{31}{5}$

Step 3.1.3

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

$\frac{2 y \cdot 5}{5} - \frac{12 y}{5} = \frac{14}{5}$

$x = - \frac{3 y}{5} - \frac{31}{5}$

Step 3.1.4

Combine the numerators over the common denominator.

$\frac{2 y \cdot 5 - 12 y}{5} = \frac{14}{5}$

$x = - \frac{3 y}{5} - \frac{31}{5}$

Step 3.1.5

Simplify the numerator.

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Step 3.1.5.1

Factor $2 y$ out of $2 y \cdot 5 - 12 y$ .

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Step 3.1.5.1.1

Factor $2 y$ out of $2 y \cdot 5$ .

$\frac{2 y (5) - 12 y}{5} = \frac{14}{5}$

$x = - \frac{3 y}{5} - \frac{31}{5}$

Step 3.1.5.1.2

Factor $2 y$ out of $- 12 y$ .

$\frac{2 y (5) + 2 y (- 6)}{5} = \frac{14}{5}$

$x = - \frac{3 y}{5} - \frac{31}{5}$

Step 3.1.5.1.3

Factor $2 y$ out of $2 y (5) + 2 y (- 6)$ .

$\frac{2 y (5 - 6)}{5} = \frac{14}{5}$

$x = - \frac{3 y}{5} - \frac{31}{5}$

$\frac{2 y (5 - 6)}{5} = \frac{14}{5}$

$x = - \frac{3 y}{5} - \frac{31}{5}$

Step 3.1.5.2

Subtract $6$ from $5$ .

$\frac{2 y \cdot - 1}{5} = \frac{14}{5}$

$x = - \frac{3 y}{5} - \frac{31}{5}$

Step 3.1.5.3

Multiply $- 1$ by $2$ .

$\frac{- 2 y}{5} = \frac{14}{5}$

$x = - \frac{3 y}{5} - \frac{31}{5}$

$\frac{- 2 y}{5} = \frac{14}{5}$

$x = - \frac{3 y}{5} - \frac{31}{5}$

Step 3.1.6

Move the negative in front of the fraction.

$- \frac{2 y}{5} = \frac{14}{5}$

$x = - \frac{3 y}{5} - \frac{31}{5}$

$- \frac{2 y}{5} = \frac{14}{5}$

$x = - \frac{3 y}{5} - \frac{31}{5}$

Step 3.2

Since the expression on each side of the equation has the same denominator, the numerators must be equal.

$- 2 y = 14$

$x = - \frac{3 y}{5} - \frac{31}{5}$

Step 3.3

Divide each term in $- 2 y = 14$ by $- 2$ and simplify.

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Step 3.3.1

Divide each term in $- 2 y = 14$ by $- 2$ .

$\frac{- 2 y}{- 2} = \frac{14}{- 2}$

$x = - \frac{3 y}{5} - \frac{31}{5}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $- 2$ .

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Step 3.3.2.1.1

Cancel the common factor.

$\frac{- 2 y}{- 2} = \frac{14}{- 2}$

$x = - \frac{3 y}{5} - \frac{31}{5}$

Step 3.3.2.1.2

Divide $y$ by $1$ .

$y = \frac{14}{- 2}$

$x = - \frac{3 y}{5} - \frac{31}{5}$

$y = \frac{14}{- 2}$

$x = - \frac{3 y}{5} - \frac{31}{5}$

$y = \frac{14}{- 2}$

$x = - \frac{3 y}{5} - \frac{31}{5}$

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

Divide $14$ by $- 2$ .

$y = - 7$

$x = - \frac{3 y}{5} - \frac{31}{5}$

$y = - 7$

$x = - \frac{3 y}{5} - \frac{31}{5}$

$y = - 7$

$x = - \frac{3 y}{5} - \frac{31}{5}$

$y = - 7$

$x = - \frac{3 y}{5} - \frac{31}{5}$

Step 4

Replace all occurrences of $y$ with $- 7$ in each equation.

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Step 4.1

Replace all occurrences of $y$ in $x = - \frac{3 y}{5} - \frac{31}{5}$ with $- 7$ .

$x = - \frac{3 (- 7)}{5} - \frac{31}{5}$

$y = - 7$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $- \frac{3 (- 7)}{5} - \frac{31}{5}$ .

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Step 4.2.1.1

Combine the numerators over the common denominator.

$x = \frac{- 3 \cdot - 7 - 31}{5}$

$y = - 7$

Step 4.2.1.2

Simplify the expression.

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Step 4.2.1.2.1

Multiply $- 3$ by $- 7$ .

$x = \frac{21 - 31}{5}$

$y = - 7$

Step 4.2.1.2.2

Subtract $31$ from $21$ .

$x = \frac{- 10}{5}$

$y = - 7$

Step 4.2.1.2.3

Divide $- 10$ by $5$ .

$x = - 2$

$y = - 7$

$x = - 2$

$y = - 7$

$x = - 2$

$y = - 7$

$x = - 2$

$y = - 7$

$x = - 2$

$y = - 7$

Step 5

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

$(- 2, - 7)$

Step 6

The result can be shown in multiple forms.

Point Form:

$(- 2, - 7)$

Equation Form:

$x = - 2, y = - 7$

Step 7