Algebra Examples

$x + y = 2$ $y = - \frac{1}{3} x - 2$

Step 1

Subtract $y$ from both sides of the equation.

$x = 2 - y$

$y = - \frac{1}{3} x - 2$

Step 2

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

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Step 2.1

Replace all occurrences of $x$ in $y = - \frac{1}{3} x - 2$ with $2 - y$ .

$y = - \frac{1}{3} \cdot (2 - y) - 2$

$x = 2 - y$

Step 2.2

Simplify the right side.

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Step 2.2.1

Simplify $- \frac{1}{3} \cdot (2 - y) - 2$ .

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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.

$y = - \frac{1}{3} \cdot 2 - \frac{1}{3} \cdot (- y) - 2$

$x = 2 - y$

Step 2.2.1.1.2

Multiply $- \frac{1}{3} \cdot 2$ .

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Step 2.2.1.1.2.1

Multiply $2$ by $- 1$ .

$y = - 2 (\frac{1}{3}) - \frac{1}{3} \cdot (- y) - 2$

$x = 2 - y$

Step 2.2.1.1.2.2

Combine $- 2$ and $\frac{1}{3}$ .

$y = \frac{- 2}{3} - \frac{1}{3} \cdot (- y) - 2$

$x = 2 - y$

$y = \frac{- 2}{3} - \frac{1}{3} \cdot (- y) - 2$

$x = 2 - y$

Step 2.2.1.1.3

Multiply $- \frac{1}{3} (- y)$ .

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Step 2.2.1.1.3.1

Multiply $- 1$ by $- 1$ .

$y = \frac{- 2}{3} + 1 (\frac{1}{3}) \cdot y - 2$

$x = 2 - y$

Step 2.2.1.1.3.2

Multiply $\frac{1}{3}$ by $1$ .

$y = \frac{- 2}{3} + \frac{1}{3} y - 2$

$x = 2 - y$

Step 2.2.1.1.3.3

Combine $\frac{1}{3}$ and $y$ .

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

$x = 2 - y$

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

$x = 2 - y$

Step 2.2.1.1.4

Move the negative in front of the fraction.

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

$x = 2 - y$

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

$x = 2 - y$

Step 2.2.1.2

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

$y = \frac{y}{3} - \frac{2}{3} - 2 \cdot \frac{3}{3}$

$x = 2 - y$

Step 2.2.1.3

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

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

$x = 2 - y$

Step 2.2.1.4

Combine the numerators over the common denominator.

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

$x = 2 - y$

Step 2.2.1.5

Simplify the numerator.

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Step 2.2.1.5.1

Multiply $- 2$ by $3$ .

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

$x = 2 - y$

Step 2.2.1.5.2

Subtract $6$ from $- 2$ .

$y = \frac{y}{3} + \frac{- 8}{3}$

$x = 2 - y$

$y = \frac{y}{3} + \frac{- 8}{3}$

$x = 2 - y$

Step 2.2.1.6

Move the negative in front of the fraction.

$y = \frac{y}{3} - \frac{8}{3}$

$x = 2 - y$

$y = \frac{y}{3} - \frac{8}{3}$

$x = 2 - y$

$y = \frac{y}{3} - \frac{8}{3}$

$x = 2 - y$

$y = \frac{y}{3} - \frac{8}{3}$

$x = 2 - y$

Step 3

Solve for $y$ in $y = \frac{y}{3} - \frac{8}{3}$ .

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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 $\frac{y}{3}$ from both sides of the equation.

$y - \frac{y}{3} = - \frac{8}{3}$

$x = 2 - y$

Step 3.1.2

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

$y \cdot \frac{3}{3} - \frac{y}{3} = - \frac{8}{3}$

$x = 2 - y$

Step 3.1.3

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

$\frac{y \cdot 3}{3} - \frac{y}{3} = - \frac{8}{3}$

$x = 2 - y$

Step 3.1.4

Combine the numerators over the common denominator.

$\frac{y \cdot 3 - y}{3} = - \frac{8}{3}$

$x = 2 - y$

Step 3.1.5

Simplify the numerator.

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Step 3.1.5.1

Move $3$ to the left of $y$ .

$\frac{3 \cdot y - y}{3} = - \frac{8}{3}$

$x = 2 - y$

Step 3.1.5.2

Subtract $y$ from $3 y$ .

$\frac{2 y}{3} = - \frac{8}{3}$

$x = 2 - y$

$\frac{2 y}{3} = - \frac{8}{3}$

$x = 2 - y$

$\frac{2 y}{3} = - \frac{8}{3}$

$x = 2 - y$

Step 3.2

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

$2 y = - 8$

$x = 2 - y$

Step 3.3

Divide each term in $2 y = - 8$ by $2$ and simplify.

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Step 3.3.1

Divide each term in $2 y = - 8$ by $2$ .

$\frac{2 y}{2} = \frac{- 8}{2}$

$x = 2 - y$

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.

$\frac{2 y}{2} = \frac{- 8}{2}$

$x = 2 - y$

Step 3.3.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 8}{2}$

$x = 2 - y$

$y = \frac{- 8}{2}$

$x = 2 - y$

$y = \frac{- 8}{2}$

$x = 2 - y$

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

Divide $- 8$ by $2$ .

$y = - 4$

$x = 2 - y$

$y = - 4$

$x = 2 - y$

$y = - 4$

$x = 2 - y$

$y = - 4$

$x = 2 - y$

Step 4

Replace all occurrences of $y$ with $- 4$ in each equation.

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Step 4.1

Replace all occurrences of $y$ in $x = 2 - y$ with $- 4$ .

$x = 2 - (- 4)$

$y = - 4$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $2 - (- 4)$ .

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Step 4.2.1.1

Multiply $- 1$ by $- 4$ .

$x = 2 + 4$

$y = - 4$

Step 4.2.1.2

Add $2$ and $4$ .

$x = 6$

$y = - 4$

$x = 6$

$y = - 4$

$x = 6$

$y = - 4$

$x = 6$

$y = - 4$

Step 5

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

$(6, - 4)$

Step 6

The result can be shown in multiple forms.

Point Form:

$(6, - 4)$

Equation Form:

$x = 6, y = - 4$

Step 7