Algebra Examples

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

Step 1

Solve for $x$ in $2 x + 7 y = - 5$ .

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Step 1.1

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

$2 x = - 5 - 7 y$

$- x - 6 y = 10$

Step 1.2

Divide each term in $2 x = - 5 - 7 y$ by $2$ and simplify.

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Step 1.2.1

Divide each term in $2 x = - 5 - 7 y$ by $2$ .

$\frac{2 x}{2} = \frac{- 5}{2} + \frac{- 7 y}{2}$

$- x - 6 y = 10$

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.

$\frac{2 x}{2} = \frac{- 5}{2} + \frac{- 7 y}{2}$

$- x - 6 y = 10$

Step 1.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 5}{2} + \frac{- 7 y}{2}$

$- x - 6 y = 10$

$x = \frac{- 5}{2} + \frac{- 7 y}{2}$

$- x - 6 y = 10$

$x = \frac{- 5}{2} + \frac{- 7 y}{2}$

$- x - 6 y = 10$

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

Move the negative in front of the fraction.

$x = - \frac{5}{2} + \frac{- 7 y}{2}$

$- x - 6 y = 10$

Step 1.2.3.1.2

Move the negative in front of the fraction.

$x = - \frac{5}{2} - \frac{7 y}{2}$

$- x - 6 y = 10$

$x = - \frac{5}{2} - \frac{7 y}{2}$

$- x - 6 y = 10$

$x = - \frac{5}{2} - \frac{7 y}{2}$

$- x - 6 y = 10$

$x = - \frac{5}{2} - \frac{7 y}{2}$

$- x - 6 y = 10$

$x = - \frac{5}{2} - \frac{7 y}{2}$

$- x - 6 y = 10$

Step 2

Replace all occurrences of $x$ with $- \frac{5}{2} - \frac{7 y}{2}$ in each equation.

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Step 2.1

Replace all occurrences of $x$ in $- x - 6 y = 10$ with $- \frac{5}{2} - \frac{7 y}{2}$ .

$- (- \frac{5}{2} - \frac{7 y}{2}) - 6 y = 10$

$x = - \frac{5}{2} - \frac{7 y}{2}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $- (- \frac{5}{2} - \frac{7 y}{2}) - 6 y$ .

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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.

$\frac{5}{2} + \frac{7 y}{2} - 6 y = 10$

$x = - \frac{5}{2} - \frac{7 y}{2}$

Step 2.2.1.1.2

Multiply $- - \frac{5}{2}$ .

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Step 2.2.1.1.2.1

Multiply $- 1$ by $- 1$ .

$1 (\frac{5}{2}) + \frac{7 y}{2} - 6 y = 10$

$x = - \frac{5}{2} - \frac{7 y}{2}$

Step 2.2.1.1.2.2

Multiply $\frac{5}{2}$ by $1$ .

$\frac{5}{2} + \frac{7 y}{2} - 6 y = 10$

$x = - \frac{5}{2} - \frac{7 y}{2}$

$\frac{5}{2} + \frac{7 y}{2} - 6 y = 10$

$x = - \frac{5}{2} - \frac{7 y}{2}$

Step 2.2.1.1.3

Multiply $- - \frac{7 y}{2}$ .

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Step 2.2.1.1.3.1

Multiply $- 1$ by $- 1$ .

$\frac{5}{2} + 1 (\frac{7 y}{2}) - 6 y = 10$

$x = - \frac{5}{2} - \frac{7 y}{2}$

Step 2.2.1.1.3.2

Multiply $\frac{7 y}{2}$ by $1$ .

$\frac{5}{2} + \frac{7 y}{2} - 6 y = 10$

$x = - \frac{5}{2} - \frac{7 y}{2}$

$\frac{5}{2} + \frac{7 y}{2} - 6 y = 10$

$x = - \frac{5}{2} - \frac{7 y}{2}$

$\frac{5}{2} + \frac{7 y}{2} - 6 y = 10$

$x = - \frac{5}{2} - \frac{7 y}{2}$

Step 2.2.1.2

To write $- 6 y$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{5}{2} + \frac{7 y}{2} - 6 y \cdot \frac{2}{2} = 10$

$x = - \frac{5}{2} - \frac{7 y}{2}$

Step 2.2.1.3

Combine $- 6 y$ and $\frac{2}{2}$ .

$\frac{5}{2} + \frac{7 y}{2} + \frac{- 6 y \cdot 2}{2} = 10$

$x = - \frac{5}{2} - \frac{7 y}{2}$

Step 2.2.1.4

Combine the numerators over the common denominator.

$\frac{5}{2} + \frac{7 y - 6 y \cdot 2}{2} = 10$

$x = - \frac{5}{2} - \frac{7 y}{2}$

Step 2.2.1.5

Combine the numerators over the common denominator.

$\frac{5 + 7 y - 6 y \cdot 2}{2} = 10$

$x = - \frac{5}{2} - \frac{7 y}{2}$

Step 2.2.1.6

Multiply $2$ by $- 6$ .

$\frac{5 + 7 y - 12 y}{2} = 10$

$x = - \frac{5}{2} - \frac{7 y}{2}$

Step 2.2.1.7

Subtract $12 y$ from $7 y$ .

$\frac{5 - 5 y}{2} = 10$

$x = - \frac{5}{2} - \frac{7 y}{2}$

Step 2.2.1.8

Factor $5$ out of $5 - 5 y$ .

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Step 2.2.1.8.1

Factor $5$ out of $5$ .

$\frac{5 (1) - 5 y}{2} = 10$

$x = - \frac{5}{2} - \frac{7 y}{2}$

Step 2.2.1.8.2

Factor $5$ out of $- 5 y$ .

$\frac{5 (1) + 5 (- y)}{2} = 10$

$x = - \frac{5}{2} - \frac{7 y}{2}$

Step 2.2.1.8.3

Factor $5$ out of $5 (1) + 5 (- y)$ .

$\frac{5 (1 - y)}{2} = 10$

$x = - \frac{5}{2} - \frac{7 y}{2}$

$\frac{5 (1 - y)}{2} = 10$

$x = - \frac{5}{2} - \frac{7 y}{2}$

$\frac{5 (1 - y)}{2} = 10$

$x = - \frac{5}{2} - \frac{7 y}{2}$

$\frac{5 (1 - y)}{2} = 10$

$x = - \frac{5}{2} - \frac{7 y}{2}$

$\frac{5 (1 - y)}{2} = 10$

$x = - \frac{5}{2} - \frac{7 y}{2}$

Step 3

Solve for $y$ in $\frac{5 (1 - y)}{2} = 10$ .

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Step 3.1

Multiply both sides by $2$ .

$\frac{5 (1 - y)}{2} \cdot 2 = 10 \cdot 2$

$x = - \frac{5}{2} - \frac{7 y}{2}$

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 $\frac{5 (1 - y)}{2} \cdot 2$ .

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Step 3.2.1.1.1

Cancel the common factor of $2$ .

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Step 3.2.1.1.1.1

Cancel the common factor.

$\frac{5 (1 - y)}{2} \cdot 2 = 10 \cdot 2$

$x = - \frac{5}{2} - \frac{7 y}{2}$

Step 3.2.1.1.1.2

Rewrite the expression.

$5 (1 - y) = 10 \cdot 2$

$x = - \frac{5}{2} - \frac{7 y}{2}$

$5 (1 - y) = 10 \cdot 2$

$x = - \frac{5}{2} - \frac{7 y}{2}$

Step 3.2.1.1.2

Apply the distributive property.

$5 \cdot 1 + 5 (- y) = 10 \cdot 2$

$x = - \frac{5}{2} - \frac{7 y}{2}$

Step 3.2.1.1.3

Simplify the expression.

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Step 3.2.1.1.3.1

Multiply $5$ by $1$ .

$5 + 5 (- y) = 10 \cdot 2$

$x = - \frac{5}{2} - \frac{7 y}{2}$

Step 3.2.1.1.3.2

Multiply $- 1$ by $5$ .

$5 - 5 y = 10 \cdot 2$

$x = - \frac{5}{2} - \frac{7 y}{2}$

Step 3.2.1.1.3.3

Reorder $5$ and $- 5 y$ .

$- 5 y + 5 = 10 \cdot 2$

$x = - \frac{5}{2} - \frac{7 y}{2}$

$- 5 y + 5 = 10 \cdot 2$

$x = - \frac{5}{2} - \frac{7 y}{2}$

$- 5 y + 5 = 10 \cdot 2$

$x = - \frac{5}{2} - \frac{7 y}{2}$

$- 5 y + 5 = 10 \cdot 2$

$x = - \frac{5}{2} - \frac{7 y}{2}$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Multiply $10$ by $2$ .

$- 5 y + 5 = 20$

$x = - \frac{5}{2} - \frac{7 y}{2}$

$- 5 y + 5 = 20$

$x = - \frac{5}{2} - \frac{7 y}{2}$

$- 5 y + 5 = 20$

$x = - \frac{5}{2} - \frac{7 y}{2}$

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 $5$ from both sides of the equation.

$- 5 y = 20 - 5$

$x = - \frac{5}{2} - \frac{7 y}{2}$

Step 3.3.1.2

Subtract $5$ from $20$ .

$- 5 y = 15$

$x = - \frac{5}{2} - \frac{7 y}{2}$

$- 5 y = 15$

$x = - \frac{5}{2} - \frac{7 y}{2}$

Step 3.3.2

Divide each term in $- 5 y = 15$ by $- 5$ and simplify.

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Step 3.3.2.1

Divide each term in $- 5 y = 15$ by $- 5$ .

$\frac{- 5 y}{- 5} = \frac{15}{- 5}$

$x = - \frac{5}{2} - \frac{7 y}{2}$

Step 3.3.2.2

Simplify the left side.

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Step 3.3.2.2.1

Cancel the common factor of $- 5$ .

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Step 3.3.2.2.1.1

Cancel the common factor.

$\frac{- 5 y}{- 5} = \frac{15}{- 5}$

$x = - \frac{5}{2} - \frac{7 y}{2}$

Step 3.3.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{15}{- 5}$

$x = - \frac{5}{2} - \frac{7 y}{2}$

$y = \frac{15}{- 5}$

$x = - \frac{5}{2} - \frac{7 y}{2}$

$y = \frac{15}{- 5}$

$x = - \frac{5}{2} - \frac{7 y}{2}$

Step 3.3.2.3

Simplify the right side.

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Step 3.3.2.3.1

Divide $15$ by $- 5$ .

$y = - 3$

$x = - \frac{5}{2} - \frac{7 y}{2}$

$y = - 3$

$x = - \frac{5}{2} - \frac{7 y}{2}$

$y = - 3$

$x = - \frac{5}{2} - \frac{7 y}{2}$

$y = - 3$

$x = - \frac{5}{2} - \frac{7 y}{2}$

$y = - 3$

$x = - \frac{5}{2} - \frac{7 y}{2}$

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 $x = - \frac{5}{2} - \frac{7 y}{2}$ with $- 3$ .

$x = - \frac{5}{2} - \frac{7 (- 3)}{2}$

$y = - 3$

Step 4.2

Simplify the right side.

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Step 4.2.1

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

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Step 4.2.1.1

Combine the numerators over the common denominator.

$x = \frac{- 5 - 7 \cdot - 3}{2}$

$y = - 3$

Step 4.2.1.2

Simplify the expression.

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Step 4.2.1.2.1

Multiply $- 7$ by $- 3$ .

$x = \frac{- 5 + 21}{2}$

$y = - 3$

Step 4.2.1.2.2

Add $- 5$ and $21$ .

$x = \frac{16}{2}$

$y = - 3$

Step 4.2.1.2.3

Divide $16$ by $2$ .

$x = 8$

$y = - 3$

$x = 8$

$y = - 3$

$x = 8$

$y = - 3$

$x = 8$

$y = - 3$

$x = 8$

$y = - 3$

Step 5

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

$(8, - 3)$

Step 6

The result can be shown in multiple forms.

Point Form:

$(8, - 3)$

Equation Form:

$x = 8, y = - 3$

Step 7