Algebra Examples

$\frac{1}{4} x - \frac{1}{2} y = 3$ $\frac{1}{2} x - \frac{3}{5} y = - 2$

Step 1

Solve for $x$ in $\frac{1}{4} x - \frac{1}{2} y = 3$ .

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Step 1.1

Simplify each term.

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Step 1.1.1

Combine $\frac{1}{4}$ and $x$ .

$\frac{x}{4} - \frac{1}{2} y = 3$

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

Step 1.1.2

Combine $y$ and $\frac{1}{2}$ .

$\frac{x}{4} - \frac{y}{2} = 3$

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

$\frac{x}{4} - \frac{y}{2} = 3$

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

Step 1.2

Add $\frac{y}{2}$ to both sides of the equation.

$\frac{x}{4} = 3 + \frac{y}{2}$

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

Step 1.3

Multiply both sides of the equation by $4$ .

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

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

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 $4$ .

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Step 1.4.1.1.1

Cancel the common factor.

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

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

Step 1.4.1.1.2

Rewrite the expression.

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

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

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

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

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

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

Step 1.4.2

Simplify the right side.

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Step 1.4.2.1

Simplify $4 (3 + \frac{y}{2})$ .

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Step 1.4.2.1.1

Apply the distributive property.

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

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

Step 1.4.2.1.2

Multiply $4$ by $3$ .

$x = 12 + 4 (\frac{y}{2})$

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

Step 1.4.2.1.3

Cancel the common factor of $2$ .

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Step 1.4.2.1.3.1

Factor $2$ out of $4$ .

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

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

Step 1.4.2.1.3.2

Cancel the common factor.

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

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

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

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

$x = 2 y + 12$

Step 2.2.1.1.5

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

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

$x = 2 y + 12$

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

$x = 2 y + 12$

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

Subtract $3 y$ from $y \cdot 5$ .

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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

Subtract $3 y$ from $5 \cdot y$ .

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

$x = 2 y + 12$

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

$x = 2 y + 12$

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

$x = 2 y + 12$

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

$x = 2 y + 12$

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

$x = 2 y + 12$

Step 3

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

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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

Subtract $6$ from both sides of the equation.

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

$x = 2 y + 12$

Step 3.1.2

Subtract $6$ from $- 2$ .

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

$x = 2 y + 12$

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

$x = 2 y + 12$

Step 3.2

Multiply both sides of the equation by $\frac{5}{2}$ .

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

$x = 2 y + 12$

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 $\frac{5}{2} \cdot \frac{2 y}{5}$ .

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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.

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

$x = 2 y + 12$

Step 3.3.1.1.1.2

Rewrite the expression.

$\frac{1}{2} \cdot (2 y) = \frac{5}{2} \cdot - 8$

$x = 2 y + 12$

$\frac{1}{2} \cdot (2 y) = \frac{5}{2} \cdot - 8$

$x = 2 y + 12$

Step 3.3.1.1.2

Cancel the common factor of $2$ .

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Step 3.3.1.1.2.1

Factor $2$ out of $2 y$ .

$\frac{1}{2} \cdot (2 (y)) = \frac{5}{2} \cdot - 8$

$x = 2 y + 12$

Step 3.3.1.1.2.2

Cancel the common factor.

$\frac{1}{2} \cdot (2 y) = \frac{5}{2} \cdot - 8$

$x = 2 y + 12$

Step 3.3.1.1.2.3

Rewrite the expression.

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

$x = 2 y + 12$

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

$x = 2 y + 12$

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

$x = 2 y + 12$

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

$x = 2 y + 12$

Step 3.3.2

Simplify the right side.

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Step 3.3.2.1

Simplify $\frac{5}{2} \cdot - 8$ .

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Step 3.3.2.1.1

Cancel the common factor of $2$ .

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Step 3.3.2.1.1.1

Factor $2$ out of $- 8$ .

$y = \frac{5}{2} \cdot (2 (- 4))$

$x = 2 y + 12$

Step 3.3.2.1.1.2

Cancel the common factor.

$y = \frac{5}{2} \cdot (2 \cdot - 4)$

$x = 2 y + 12$

Step 3.3.2.1.1.3

Rewrite the expression.

$y = 5 \cdot - 4$

$x = 2 y + 12$

$y = 5 \cdot - 4$

$x = 2 y + 12$

Step 3.3.2.1.2

Multiply $5$ by $- 4$ .

$y = - 20$

$x = 2 y + 12$

$y = - 20$

$x = 2 y + 12$

$y = - 20$

$x = 2 y + 12$

$y = - 20$

$x = 2 y + 12$

$y = - 20$

$x = 2 y + 12$

Step 4

Replace all occurrences of $y$ with $- 20$ in each equation.

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Step 4.1

Replace all occurrences of $y$ in $x = 2 y + 12$ with $- 20$ .

$x = 2 (- 20) + 12$

$y = - 20$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $2 (- 20) + 12$ .

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Step 4.2.1.1

Multiply $2$ by $- 20$ .

$x = - 40 + 12$

$y = - 20$

Step 4.2.1.2

Add $- 40$ and $12$ .

$x = - 28$

$y = - 20$

$x = - 28$

$y = - 20$

$x = - 28$

$y = - 20$

$x = - 28$

$y = - 20$

Step 5

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

$(- 28, - 20)$

Step 6

The result can be shown in multiple forms.

Point Form:

$(- 28, - 20)$

Equation Form:

$x = - 28, y = - 20$

Step 7