Algebra Examples

$4 x - \frac{1}{3} y = 8$ $\frac{2}{3} x + \frac{1}{3} y = \frac{11}{3}$

Step 1

Solve for $x$ in $4 x - \frac{1}{3} y = 8$ .

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Step 1.1

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

$4 x - \frac{y}{3} = 8$

$\frac{2}{3} x + \frac{1}{3} y = \frac{11}{3}$

Step 1.2

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

$4 x = 8 + \frac{y}{3}$

$\frac{2}{3} x + \frac{1}{3} y = \frac{11}{3}$

Step 1.3

Divide each term in $4 x = 8 + \frac{y}{3}$ by $4$ and simplify.

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Step 1.3.1

Divide each term in $4 x = 8 + \frac{y}{3}$ by $4$ .

$\frac{4 x}{4} = \frac{8}{4} + \frac{\frac{y}{3}}{4}$

$\frac{2}{3} x + \frac{1}{3} y = \frac{11}{3}$

Step 1.3.2

Simplify the left side.

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Step 1.3.2.1

Cancel the common factor of $4$ .

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Step 1.3.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{8}{4} + \frac{\frac{y}{3}}{4}$

$\frac{2}{3} x + \frac{1}{3} y = \frac{11}{3}$

Step 1.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{8}{4} + \frac{\frac{y}{3}}{4}$

$\frac{2}{3} x + \frac{1}{3} y = \frac{11}{3}$

$x = \frac{8}{4} + \frac{\frac{y}{3}}{4}$

$\frac{2}{3} x + \frac{1}{3} y = \frac{11}{3}$

$x = \frac{8}{4} + \frac{\frac{y}{3}}{4}$

$\frac{2}{3} x + \frac{1}{3} y = \frac{11}{3}$

Step 1.3.3

Simplify the right side.

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Step 1.3.3.1

Simplify each term.

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Step 1.3.3.1.1

Divide $8$ by $4$ .

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

$\frac{2}{3} x + \frac{1}{3} y = \frac{11}{3}$

Step 1.3.3.1.2

Multiply the numerator by the reciprocal of the denominator.

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

$\frac{2}{3} x + \frac{1}{3} y = \frac{11}{3}$

Step 1.3.3.1.3

Multiply $\frac{y}{3} \cdot \frac{1}{4}$ .

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Step 1.3.3.1.3.1

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

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

$\frac{2}{3} x + \frac{1}{3} y = \frac{11}{3}$

Step 1.3.3.1.3.2

Multiply $3$ by $4$ .

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

$\frac{2}{3} x + \frac{1}{3} y = \frac{11}{3}$

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

$\frac{2}{3} x + \frac{1}{3} y = \frac{11}{3}$

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

$\frac{2}{3} x + \frac{1}{3} y = \frac{11}{3}$

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

$\frac{2}{3} x + \frac{1}{3} y = \frac{11}{3}$

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

$\frac{2}{3} x + \frac{1}{3} y = \frac{11}{3}$

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

$\frac{2}{3} x + \frac{1}{3} y = \frac{11}{3}$

Step 2

Replace all occurrences of $x$ with $2 + \frac{y}{12}$ in each equation.

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Step 2.1

Replace all occurrences of $x$ in $\frac{2}{3} x + \frac{1}{3} y = \frac{11}{3}$ with $2 + \frac{y}{12}$ .

$\frac{2}{3} \cdot (2 + \frac{y}{12}) + \frac{1}{3} y = \frac{11}{3}$

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

Step 2.2

Simplify the left side.

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Step 2.2.1

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

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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.

$\frac{2}{3} \cdot 2 + \frac{2}{3} \cdot \frac{y}{12} + \frac{1}{3} y = \frac{11}{3}$

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

Step 2.2.1.1.2

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

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Step 2.2.1.1.2.1

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

$\frac{2 \cdot 2}{3} + \frac{2}{3} \cdot \frac{y}{12} + \frac{1}{3} y = \frac{11}{3}$

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

Step 2.2.1.1.2.2

Multiply $2$ by $2$ .

$\frac{4}{3} + \frac{2}{3} \cdot \frac{y}{12} + \frac{1}{3} y = \frac{11}{3}$

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

$\frac{4}{3} + \frac{2}{3} \cdot \frac{y}{12} + \frac{1}{3} y = \frac{11}{3}$

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

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 $12$ .

$\frac{4}{3} + \frac{2}{3} \cdot \frac{y}{2 (6)} + \frac{1}{3} y = \frac{11}{3}$

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

Step 2.2.1.1.3.2

Cancel the common factor.

$\frac{4}{3} + \frac{2}{3} \cdot \frac{y}{2 \cdot 6} + \frac{1}{3} y = \frac{11}{3}$

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

Step 2.2.1.1.3.3

Rewrite the expression.

$\frac{4}{3} + \frac{1}{3} \cdot \frac{y}{6} + \frac{1}{3} y = \frac{11}{3}$

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

$\frac{4}{3} + \frac{1}{3} \cdot \frac{y}{6} + \frac{1}{3} y = \frac{11}{3}$

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

Step 2.2.1.1.4

Multiply $\frac{1}{3}$ by $\frac{y}{6}$ .

$\frac{4}{3} + \frac{y}{3 \cdot 6} + \frac{1}{3} y = \frac{11}{3}$

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

Step 2.2.1.1.5

Multiply $3$ by $6$ .

$\frac{4}{3} + \frac{y}{18} + \frac{1}{3} y = \frac{11}{3}$

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

Step 2.2.1.1.6

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

$\frac{4}{3} + \frac{y}{18} + \frac{y}{3} = \frac{11}{3}$

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

$\frac{4}{3} + \frac{y}{18} + \frac{y}{3} = \frac{11}{3}$

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

Step 2.2.1.2

To write $\frac{y}{3}$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$\frac{4}{3} + \frac{y}{18} + \frac{y}{3} \cdot \frac{6}{6} = \frac{11}{3}$

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

Step 2.2.1.3

Write each expression with a common denominator of $18$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.2.1.3.1

Multiply $\frac{y}{3}$ by $\frac{6}{6}$ .

$\frac{4}{3} + \frac{y}{18} + \frac{y \cdot 6}{3 \cdot 6} = \frac{11}{3}$

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

Step 2.2.1.3.2

Multiply $3$ by $6$ .

$\frac{4}{3} + \frac{y}{18} + \frac{y \cdot 6}{18} = \frac{11}{3}$

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

$\frac{4}{3} + \frac{y}{18} + \frac{y \cdot 6}{18} = \frac{11}{3}$

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

Step 2.2.1.4

Combine the numerators over the common denominator.

$\frac{4}{3} + \frac{y + y \cdot 6}{18} = \frac{11}{3}$

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

Step 2.2.1.5

Add $y$ and $y \cdot 6$ .

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Step 2.2.1.5.1

Reorder $y$ and $6$ .

$\frac{4}{3} + \frac{y + 6 \cdot y}{18} = \frac{11}{3}$

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

Step 2.2.1.5.2

Add $y$ and $6 \cdot y$ .

$\frac{4}{3} + \frac{7 y}{18} = \frac{11}{3}$

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

$\frac{4}{3} + \frac{7 y}{18} = \frac{11}{3}$

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

$\frac{4}{3} + \frac{7 y}{18} = \frac{11}{3}$

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

$\frac{4}{3} + \frac{7 y}{18} = \frac{11}{3}$

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

$\frac{4}{3} + \frac{7 y}{18} = \frac{11}{3}$

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

Step 3

Solve for $y$ in $\frac{4}{3} + \frac{7 y}{18} = \frac{11}{3}$ .

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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 $\frac{4}{3}$ from both sides of the equation.

$\frac{7 y}{18} = \frac{11}{3} - \frac{4}{3}$

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

Step 3.1.2

Combine the numerators over the common denominator.

$\frac{7 y}{18} = \frac{11 - 4}{3}$

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

Step 3.1.3

Subtract $4$ from $11$ .

$\frac{7 y}{18} = \frac{7}{3}$

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

$\frac{7 y}{18} = \frac{7}{3}$

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

Step 3.2

Multiply both sides of the equation by $\frac{18}{7}$ .

$\frac{18}{7} \cdot \frac{7 y}{18} = \frac{18}{7} \cdot \frac{7}{3}$

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

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 $\frac{18}{7} \cdot \frac{7 y}{18}$ .

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Step 3.3.1.1.1

Cancel the common factor of $18$ .

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Step 3.3.1.1.1.1

Cancel the common factor.

$\frac{18}{7} \cdot \frac{7 y}{18} = \frac{18}{7} \cdot \frac{7}{3}$

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

Step 3.3.1.1.1.2

Rewrite the expression.

$\frac{1}{7} \cdot (7 y) = \frac{18}{7} \cdot \frac{7}{3}$

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

$\frac{1}{7} \cdot (7 y) = \frac{18}{7} \cdot \frac{7}{3}$

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

Step 3.3.1.1.2

Cancel the common factor of $7$ .

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Step 3.3.1.1.2.1

Factor $7$ out of $7 y$ .

$\frac{1}{7} \cdot (7 (y)) = \frac{18}{7} \cdot \frac{7}{3}$

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

Step 3.3.1.1.2.2

Cancel the common factor.

$\frac{1}{7} \cdot (7 y) = \frac{18}{7} \cdot \frac{7}{3}$

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

Step 3.3.1.1.2.3

Rewrite the expression.

$y = \frac{18}{7} \cdot \frac{7}{3}$

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

$y = \frac{18}{7} \cdot \frac{7}{3}$

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

$y = \frac{18}{7} \cdot \frac{7}{3}$

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

$y = \frac{18}{7} \cdot \frac{7}{3}$

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

Step 3.3.2

Simplify the right side.

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Step 3.3.2.1

Simplify $\frac{18}{7} \cdot \frac{7}{3}$ .

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Step 3.3.2.1.1

Cancel the common factor of $3$ .

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Step 3.3.2.1.1.1

Factor $3$ out of $18$ .

$y = \frac{3 (6)}{7} \cdot \frac{7}{3}$

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

Step 3.3.2.1.1.2

Cancel the common factor.

$y = \frac{3 \cdot 6}{7} \cdot \frac{7}{3}$

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

Step 3.3.2.1.1.3

Rewrite the expression.

$y = \frac{6}{7} \cdot 7$

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

$y = \frac{6}{7} \cdot 7$

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

Step 3.3.2.1.2

Cancel the common factor of $7$ .

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Step 3.3.2.1.2.1

Cancel the common factor.

$y = \frac{6}{7} \cdot 7$

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

Step 3.3.2.1.2.2

Rewrite the expression.

$y = 6$

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

$y = 6$

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

$y = 6$

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

$y = 6$

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

$y = 6$

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

$y = 6$

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

Step 4

Replace all occurrences of $y$ with $6$ in each equation.

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Step 4.1

Replace all occurrences of $y$ in $x = 2 + \frac{y}{12}$ with $6$ .

$x = 2 + \frac{6}{12}$

$y = 6$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $2 + \frac{6}{12}$ .

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Step 4.2.1.1

Cancel the common factor of $6$ and $12$ .

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Step 4.2.1.1.1

Factor $6$ out of $6$ .

$x = 2 + \frac{6 (1)}{12}$

$y = 6$

Step 4.2.1.1.2

Cancel the common factors.

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Step 4.2.1.1.2.1

Factor $6$ out of $12$ .

$x = 2 + \frac{6 \cdot 1}{6 \cdot 2}$

$y = 6$

Step 4.2.1.1.2.2

Cancel the common factor.

$x = 2 + \frac{6 \cdot 1}{6 \cdot 2}$

$y = 6$

Step 4.2.1.1.2.3

Rewrite the expression.

$x = 2 + \frac{1}{2}$

$y = 6$

$x = 2 + \frac{1}{2}$

$y = 6$

$x = 2 + \frac{1}{2}$

$y = 6$

Step 4.2.1.2

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

$x = 2 \cdot \frac{2}{2} + \frac{1}{2}$

$y = 6$

Step 4.2.1.3

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

$x = \frac{2 \cdot 2}{2} + \frac{1}{2}$

$y = 6$

Step 4.2.1.4

Combine the numerators over the common denominator.

$x = \frac{2 \cdot 2 + 1}{2}$

$y = 6$

Step 4.2.1.5

Simplify the numerator.

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Step 4.2.1.5.1

Multiply $2$ by $2$ .

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

$y = 6$

Step 4.2.1.5.2

Add $4$ and $1$ .

$x = \frac{5}{2}$

$y = 6$

$x = \frac{5}{2}$

$y = 6$

$x = \frac{5}{2}$

$y = 6$

$x = \frac{5}{2}$

$y = 6$

$x = \frac{5}{2}$

$y = 6$

Step 5

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

$(\frac{5}{2}, 6)$

Step 6

The result can be shown in multiple forms.

Point Form:

$(\frac{5}{2}, 6)$

Equation Form:

$x = \frac{5}{2}, y = 6$

Step 7