Algebra Examples

$\frac{1}{3} x + \frac{1}{2} y = 6$ $\frac{1}{4} x + \frac{1}{2} y = 3$

Step 1

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

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Step 1.1

Simplify each term.

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Step 1.1.1

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

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

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

Step 1.1.2

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

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

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

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

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

Step 1.2

Subtract $\frac{y}{2}$ from both sides of the equation.

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

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

Step 1.3

Multiply both sides of the equation by $3$ .

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

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

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 $3$ .

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Step 1.4.1.1.1

Cancel the common factor.

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

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

Step 1.4.1.1.2

Rewrite the expression.

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

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

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

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

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

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

Step 1.4.2

Simplify the right side.

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Step 1.4.2.1

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

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Step 1.4.2.1.1

Apply the distributive property.

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

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

Step 1.4.2.1.2

Multiply $3$ by $6$ .

$x = 18 + 3 (- \frac{y}{2})$

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

Step 1.4.2.1.3

Multiply $3 (- \frac{y}{2})$ .

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Step 1.4.2.1.3.1

Multiply $- 1$ by $3$ .

$x = 18 - 3 \frac{y}{2}$

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

Step 1.4.2.1.3.2

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

$x = 18 + \frac{- 3 y}{2}$

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

$x = 18 + \frac{- 3 y}{2}$

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

Step 1.4.2.1.4

Move the negative in front of the fraction.

$x = 18 - \frac{3 y}{2}$

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

$x = 18 - \frac{3 y}{2}$

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

$x = 18 - \frac{3 y}{2}$

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

$x = 18 - \frac{3 y}{2}$

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

Step 1.5

Reorder $18$ and $- \frac{3 y}{2}$ .

$x = - \frac{3 y}{2} + 18$

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

$x = - \frac{3 y}{2} + 18$

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

Step 2

Replace all occurrences of $x$ with $- \frac{3 y}{2} + 18$ in each equation.

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Step 2.1

Replace all occurrences of $x$ in $\frac{1}{4} x + \frac{1}{2} y = 3$ with $- \frac{3 y}{2} + 18$ .

$\frac{1}{4} \cdot (- \frac{3 y}{2} + 18) + \frac{1}{2} y = 3$

$x = - \frac{3 y}{2} + 18$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $\frac{1}{4} \cdot (- \frac{3 y}{2} + 18) + \frac{1}{2} y$ .

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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.

$\frac{1}{4} \cdot (- \frac{3 y}{2}) + \frac{1}{4} \cdot 18 + \frac{1}{2} y = 3$

$x = - \frac{3 y}{2} + 18$

Step 2.2.1.1.2

Multiply $\frac{1}{4} (- \frac{3 y}{2})$ .

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Step 2.2.1.1.2.1

Multiply $\frac{1}{4}$ by $\frac{3 y}{2}$ .

$- \frac{3 y}{4 \cdot 2} + \frac{1}{4} \cdot 18 + \frac{1}{2} y = 3$

$x = - \frac{3 y}{2} + 18$

Step 2.2.1.1.2.2

Multiply $4$ by $2$ .

$- \frac{3 y}{8} + \frac{1}{4} \cdot 18 + \frac{1}{2} y = 3$

$x = - \frac{3 y}{2} + 18$

$- \frac{3 y}{8} + \frac{1}{4} \cdot 18 + \frac{1}{2} y = 3$

$x = - \frac{3 y}{2} + 18$

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 $4$ .

$- \frac{3 y}{8} + \frac{1}{2 (2)} \cdot 18 + \frac{1}{2} y = 3$

$x = - \frac{3 y}{2} + 18$

Step 2.2.1.1.3.2

Factor $2$ out of $18$ .

$- \frac{3 y}{8} + \frac{1}{2 \cdot 2} \cdot (2 \cdot 9) + \frac{1}{2} y = 3$

$x = - \frac{3 y}{2} + 18$

Step 2.2.1.1.3.3

Cancel the common factor.

$- \frac{3 y}{8} + \frac{1}{2 \cdot 2} \cdot (2 \cdot 9) + \frac{1}{2} y = 3$

$x = - \frac{3 y}{2} + 18$

Step 2.2.1.1.3.4

Rewrite the expression.

$- \frac{3 y}{8} + \frac{1}{2} \cdot 9 + \frac{1}{2} y = 3$

$x = - \frac{3 y}{2} + 18$

$- \frac{3 y}{8} + \frac{1}{2} \cdot 9 + \frac{1}{2} y = 3$

$x = - \frac{3 y}{2} + 18$

Step 2.2.1.1.4

Combine $\frac{1}{2}$ and $9$ .

$- \frac{3 y}{8} + \frac{9}{2} + \frac{1}{2} y = 3$

$x = - \frac{3 y}{2} + 18$

Step 2.2.1.1.5

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

$- \frac{3 y}{8} + \frac{9}{2} + \frac{y}{2} = 3$

$x = - \frac{3 y}{2} + 18$

$- \frac{3 y}{8} + \frac{9}{2} + \frac{y}{2} = 3$

$x = - \frac{3 y}{2} + 18$

Step 2.2.1.2

To write $\frac{y}{2}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$- \frac{3 y}{8} + \frac{y}{2} \cdot \frac{4}{4} + \frac{9}{2} = 3$

$x = - \frac{3 y}{2} + 18$

Step 2.2.1.3

Write each expression with a common denominator of $8$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.2.1.3.1

Multiply $\frac{y}{2}$ by $\frac{4}{4}$ .

$- \frac{3 y}{8} + \frac{y \cdot 4}{2 \cdot 4} + \frac{9}{2} = 3$

$x = - \frac{3 y}{2} + 18$

Step 2.2.1.3.2

Multiply $2$ by $4$ .

$- \frac{3 y}{8} + \frac{y \cdot 4}{8} + \frac{9}{2} = 3$

$x = - \frac{3 y}{2} + 18$

$- \frac{3 y}{8} + \frac{y \cdot 4}{8} + \frac{9}{2} = 3$

$x = - \frac{3 y}{2} + 18$

Step 2.2.1.4

Combine the numerators over the common denominator.

$\frac{- 3 y + y \cdot 4}{8} + \frac{9}{2} = 3$

$x = - \frac{3 y}{2} + 18$

Step 2.2.1.5

Add $- 3 y$ and $y \cdot 4$ .

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Step 2.2.1.5.1

Reorder $y$ and $4$ .

$\frac{- 3 y + 4 \cdot y}{8} + \frac{9}{2} = 3$

$x = - \frac{3 y}{2} + 18$

Step 2.2.1.5.2

Add $- 3 y$ and $4 \cdot y$ .

$\frac{y}{8} + \frac{9}{2} = 3$

$x = - \frac{3 y}{2} + 18$

$\frac{y}{8} + \frac{9}{2} = 3$

$x = - \frac{3 y}{2} + 18$

$\frac{y}{8} + \frac{9}{2} = 3$

$x = - \frac{3 y}{2} + 18$

$\frac{y}{8} + \frac{9}{2} = 3$

$x = - \frac{3 y}{2} + 18$

$\frac{y}{8} + \frac{9}{2} = 3$

$x = - \frac{3 y}{2} + 18$

Step 3

Solve for $y$ in $\frac{y}{8} + \frac{9}{2} = 3$ .

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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 $\frac{9}{2}$ from both sides of the equation.

$\frac{y}{8} = 3 - \frac{9}{2}$

$x = - \frac{3 y}{2} + 18$

Step 3.1.2

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

$\frac{y}{8} = 3 \cdot \frac{2}{2} - \frac{9}{2}$

$x = - \frac{3 y}{2} + 18$

Step 3.1.3

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

$\frac{y}{8} = \frac{3 \cdot 2}{2} - \frac{9}{2}$

$x = - \frac{3 y}{2} + 18$

Step 3.1.4

Combine the numerators over the common denominator.

$\frac{y}{8} = \frac{3 \cdot 2 - 9}{2}$

$x = - \frac{3 y}{2} + 18$

Step 3.1.5

Simplify the numerator.

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Step 3.1.5.1

Multiply $3$ by $2$ .

$\frac{y}{8} = \frac{6 - 9}{2}$

$x = - \frac{3 y}{2} + 18$

Step 3.1.5.2

Subtract $9$ from $6$ .

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

$x = - \frac{3 y}{2} + 18$

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

$x = - \frac{3 y}{2} + 18$

Step 3.1.6

Move the negative in front of the fraction.

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

$x = - \frac{3 y}{2} + 18$

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

$x = - \frac{3 y}{2} + 18$

Step 3.2

Multiply both sides of the equation by $8$ .

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

$x = - \frac{3 y}{2} + 18$

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

Cancel the common factor of $8$ .

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Step 3.3.1.1.1

Cancel the common factor.

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

$x = - \frac{3 y}{2} + 18$

Step 3.3.1.1.2

Rewrite the expression.

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

$x = - \frac{3 y}{2} + 18$

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

$x = - \frac{3 y}{2} + 18$

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

$x = - \frac{3 y}{2} + 18$

Step 3.3.2

Simplify the right side.

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Step 3.3.2.1

Simplify $8 (- \frac{3}{2})$ .

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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

Move the leading negative in $- \frac{3}{2}$ into the numerator.

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

$x = - \frac{3 y}{2} + 18$

Step 3.3.2.1.1.2

Factor $2$ out of $8$ .

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

$x = - \frac{3 y}{2} + 18$

Step 3.3.2.1.1.3

Cancel the common factor.

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

$x = - \frac{3 y}{2} + 18$

Step 3.3.2.1.1.4

Rewrite the expression.

$y = 4 \cdot - 3$

$x = - \frac{3 y}{2} + 18$

$y = 4 \cdot - 3$

$x = - \frac{3 y}{2} + 18$

Step 3.3.2.1.2

Multiply $4$ by $- 3$ .

$y = - 12$

$x = - \frac{3 y}{2} + 18$

$y = - 12$

$x = - \frac{3 y}{2} + 18$

$y = - 12$

$x = - \frac{3 y}{2} + 18$

$y = - 12$

$x = - \frac{3 y}{2} + 18$

$y = - 12$

$x = - \frac{3 y}{2} + 18$

Step 4

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

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Step 4.1

Replace all occurrences of $y$ in $x = - \frac{3 y}{2} + 18$ with $- 12$ .

$x = - \frac{3 (- 12)}{2} + 18$

$y = - 12$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $- \frac{3 (- 12)}{2} + 18$ .

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Step 4.2.1.1

Simplify each term.

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Step 4.2.1.1.1

Cancel the common factor of $- 12$ and $2$ .

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Step 4.2.1.1.1.1

Factor $2$ out of $3 (- 12)$ .

$x = - \frac{2 (3 \cdot (- 6))}{2} + 18$

$y = - 12$

Step 4.2.1.1.1.2

Cancel the common factors.

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Step 4.2.1.1.1.2.1

Factor $2$ out of $2$ .

$x = - \frac{2 (3 \cdot (- 6))}{2 (1)} + 18$

$y = - 12$

Step 4.2.1.1.1.2.2

Cancel the common factor.

$x = - \frac{2 (3 \cdot (- 6))}{2 \cdot 1} + 18$

$y = - 12$

Step 4.2.1.1.1.2.3

Rewrite the expression.

$x = - \frac{3 \cdot (- 6)}{1} + 18$

$y = - 12$

Step 4.2.1.1.1.2.4

Divide $3 \cdot (- 6)$ by $1$ .

$x = - (3 \cdot (- 6)) + 18$

$y = - 12$

$x = - (3 \cdot (- 6)) + 18$

$y = - 12$

$x = - (3 \cdot (- 6)) + 18$

$y = - 12$

Step 4.2.1.1.2

Multiply $- (3 \cdot (- 6))$ .

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Step 4.2.1.1.2.1

Multiply $3$ by $- 6$ .

$x = 18 + 18$

$y = - 12$

Step 4.2.1.1.2.2

Multiply $- 1$ by $- 18$ .

$x = 18 + 18$

$y = - 12$

$x = 18 + 18$

$y = - 12$

$x = 18 + 18$

$y = - 12$

Step 4.2.1.2

Add $18$ and $18$ .

$x = 36$

$y = - 12$

$x = 36$

$y = - 12$

$x = 36$

$y = - 12$

$x = 36$

$y = - 12$

Step 5

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

$(36, - 12)$

Step 6

The result can be shown in multiple forms.

Point Form:

$(36, - 12)$

Equation Form:

$x = 36, y = - 12$

Step 7