Algebra Examples

$\frac{1}{6} x - \frac{1}{3} y = - 2$ $\frac{1}{2} x + \frac{3}{5} y = 2$

Step 1

Solve for $x$ in $\frac{1}{6} x - \frac{1}{3} y = - 2$ .

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Step 1.1

Simplify each term.

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Step 1.1.1

Combine $\frac{1}{6}$ and $x$ .

$\frac{x}{6} - \frac{1}{3} y = - 2$

$\frac{1}{2} x + \frac{3}{5} y = 2$

Step 1.1.2

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

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

$\frac{1}{2} x + \frac{3}{5} y = 2$

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

$\frac{1}{2} x + \frac{3}{5} y = 2$

Step 1.2

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

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

$\frac{1}{2} x + \frac{3}{5} y = 2$

Step 1.3

Multiply both sides of the equation by $6$ .

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

$\frac{1}{2} x + \frac{3}{5} y = 2$

Step 1.4

Simplify both sides of the equation.

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Step 1.4.1

Simplify the left side.

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Step 1.4.1.1

Cancel the common factor of $6$ .

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Step 1.4.1.1.1

Cancel the common factor.

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

$\frac{1}{2} x + \frac{3}{5} y = 2$

Step 1.4.1.1.2

Rewrite the expression.

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

$\frac{1}{2} x + \frac{3}{5} y = 2$

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

$\frac{1}{2} x + \frac{3}{5} y = 2$

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

$\frac{1}{2} x + \frac{3}{5} y = 2$

Step 1.4.2

Simplify the right side.

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Step 1.4.2.1

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

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Step 1.4.2.1.1

Apply the distributive property.

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

$\frac{1}{2} x + \frac{3}{5} y = 2$

Step 1.4.2.1.2

Multiply $6$ by $- 2$ .

$x = - 12 + 6 (\frac{y}{3})$

$\frac{1}{2} x + \frac{3}{5} y = 2$

Step 1.4.2.1.3

Cancel the common factor of $3$ .

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Step 1.4.2.1.3.1

Factor $3$ out of $6$ .

$x = - 12 + 3 (2) (\frac{y}{3})$

$\frac{1}{2} x + \frac{3}{5} y = 2$

Step 1.4.2.1.3.2

Cancel the common factor.

$x = - 12 + 3 \cdot (2 (\frac{y}{3}))$

$\frac{1}{2} x + \frac{3}{5} y = 2$

Step 1.4.2.1.3.3

Rewrite the expression.

$x = - 12 + 2 y$

$\frac{1}{2} x + \frac{3}{5} y = 2$

$x = - 12 + 2 y$

$\frac{1}{2} x + \frac{3}{5} y = 2$

$x = - 12 + 2 y$

$\frac{1}{2} x + \frac{3}{5} y = 2$

$x = - 12 + 2 y$

$\frac{1}{2} x + \frac{3}{5} y = 2$

$x = - 12 + 2 y$

$\frac{1}{2} x + \frac{3}{5} y = 2$

Step 1.5

Reorder $- 12$ and $2 y$ .

$x = 2 y - 12$

$\frac{1}{2} x + \frac{3}{5} y = 2$

$x = 2 y - 12$

$\frac{1}{2} x + \frac{3}{5} y = 2$

Step 2

Replace all occurrences of $x$ with $2 y - 12$ in each equation.

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Step 2.1

Replace all occurrences of $x$ in $\frac{1}{2} x + \frac{3}{5} y = 2$ with $2 y - 12$ .

$\frac{1}{2} \cdot (2 y - 12) + \frac{3}{5} y = 2$

$x = 2 y - 12$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $\frac{1}{2} \cdot (2 y - 12) + \frac{3}{5} y$ .

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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.

$\frac{1}{2} \cdot (2 y) + \frac{1}{2} \cdot - 12 + \frac{3}{5} y = 2$

$x = 2 y - 12$

Step 2.2.1.1.2

Cancel the common factor of $2$ .

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Step 2.2.1.1.2.1

Factor $2$ out of $2 y$ .

$\frac{1}{2} \cdot (2 (y)) + \frac{1}{2} \cdot - 12 + \frac{3}{5} y = 2$

$x = 2 y - 12$

Step 2.2.1.1.2.2

Cancel the common factor.

$\frac{1}{2} \cdot (2 y) + \frac{1}{2} \cdot - 12 + \frac{3}{5} y = 2$

$x = 2 y - 12$

Step 2.2.1.1.2.3

Rewrite the expression.

$y + \frac{1}{2} \cdot - 12 + \frac{3}{5} y = 2$

$x = 2 y - 12$

$y + \frac{1}{2} \cdot - 12 + \frac{3}{5} y = 2$

$x = 2 y - 12$

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 $- 12$ .

$y + \frac{1}{2} \cdot (2 (- 6)) + \frac{3}{5} y = 2$

$x = 2 y - 12$

Step 2.2.1.1.3.2

Cancel the common factor.

$y + \frac{1}{2} \cdot (2 \cdot - 6) + \frac{3}{5} y = 2$

$x = 2 y - 12$

Step 2.2.1.1.3.3

Rewrite the expression.

$y - 6 + \frac{3}{5} y = 2$

$x = 2 y - 12$

$y - 6 + \frac{3}{5} y = 2$

$x = 2 y - 12$

Step 2.2.1.1.4

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

$y - 6 + \frac{3 y}{5} = 2$

$x = 2 y - 12$

$y - 6 + \frac{3 y}{5} = 2$

$x = 2 y - 12$

Step 2.2.1.2

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

$y \cdot \frac{5}{5} + \frac{3 y}{5} - 6 = 2$

$x = 2 y - 12$

Step 2.2.1.3

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

$\frac{y \cdot 5}{5} + \frac{3 y}{5} - 6 = 2$

$x = 2 y - 12$

Step 2.2.1.4

Combine the numerators over the common denominator.

$\frac{y \cdot 5 + 3 y}{5} - 6 = 2$

$x = 2 y - 12$

Step 2.2.1.5

Add $y \cdot 5$ and $3 y$ .

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Step 2.2.1.5.1

Reorder $y$ and $5$ .

$\frac{5 \cdot y + 3 y}{5} - 6 = 2$

$x = 2 y - 12$

Step 2.2.1.5.2

Add $5 \cdot y$ and $3 y$ .

$\frac{8 \cdot y}{5} - 6 = 2$

$x = 2 y - 12$

$\frac{8 y}{5} - 6 = 2$

$x = 2 y - 12$

$\frac{8 y}{5} - 6 = 2$

$x = 2 y - 12$

$\frac{8 y}{5} - 6 = 2$

$x = 2 y - 12$

$\frac{8 y}{5} - 6 = 2$

$x = 2 y - 12$

Step 3

Solve for $y$ in $\frac{8 y}{5} - 6 = 2$ .

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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 $6$ to both sides of the equation.

$\frac{8 y}{5} = 2 + 6$

$x = 2 y - 12$

Step 3.1.2

Add $2$ and $6$ .

$\frac{8 y}{5} = 8$

$x = 2 y - 12$

$\frac{8 y}{5} = 8$

$x = 2 y - 12$

Step 3.2

Multiply both sides of the equation by $\frac{5}{8}$ .

$\frac{5}{8} \cdot \frac{8 y}{5} = \frac{5}{8} \cdot 8$

$x = 2 y - 12$

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

Simplify $\frac{5}{8} \cdot \frac{8 y}{5}$ .

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Step 3.3.1.1.1

Cancel the common factor of $5$ .

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Step 3.3.1.1.1.1

Cancel the common factor.

$\frac{5}{8} \cdot \frac{8 y}{5} = \frac{5}{8} \cdot 8$

$x = 2 y - 12$

Step 3.3.1.1.1.2

Rewrite the expression.

$\frac{1}{8} \cdot (8 y) = \frac{5}{8} \cdot 8$

$x = 2 y - 12$

$\frac{1}{8} \cdot (8 y) = \frac{5}{8} \cdot 8$

$x = 2 y - 12$

Step 3.3.1.1.2

Cancel the common factor of $8$ .

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Step 3.3.1.1.2.1

Factor $8$ out of $8 y$ .

$\frac{1}{8} \cdot (8 (y)) = \frac{5}{8} \cdot 8$

$x = 2 y - 12$

Step 3.3.1.1.2.2

Cancel the common factor.

$\frac{1}{8} \cdot (8 y) = \frac{5}{8} \cdot 8$

$x = 2 y - 12$

Step 3.3.1.1.2.3

Rewrite the expression.

$y = \frac{5}{8} \cdot 8$

$x = 2 y - 12$

$y = \frac{5}{8} \cdot 8$

$x = 2 y - 12$

$y = \frac{5}{8} \cdot 8$

$x = 2 y - 12$

$y = \frac{5}{8} \cdot 8$

$x = 2 y - 12$

Step 3.3.2

Simplify the right side.

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Step 3.3.2.1

Cancel the common factor of $8$ .

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Step 3.3.2.1.1

Cancel the common factor.

$y = \frac{5}{8} \cdot 8$

$x = 2 y - 12$

Step 3.3.2.1.2

Rewrite the expression.

$y = 5$

$x = 2 y - 12$

$y = 5$

$x = 2 y - 12$

$y = 5$

$x = 2 y - 12$

$y = 5$

$x = 2 y - 12$

$y = 5$

$x = 2 y - 12$

Step 4

Replace all occurrences of $y$ with $5$ in each equation.

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Step 4.1

Replace all occurrences of $y$ in $x = 2 y - 12$ with $5$ .

$x = 2 (5) - 12$

$y = 5$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $2 (5) - 12$ .

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Step 4.2.1.1

Multiply $2$ by $5$ .

$x = 10 - 12$

$y = 5$

Step 4.2.1.2

Subtract $12$ from $10$ .

$x = - 2$

$y = 5$

$x = - 2$

$y = 5$

$x = - 2$

$y = 5$

$x = - 2$

$y = 5$

Step 5

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

$(- 2, 5)$

Step 6

The result can be shown in multiple forms.

Point Form:

$(- 2, 5)$

Equation Form:

$x = - 2, y = 5$

Step 7