Algebra Examples

$3 x - \frac{y - 3}{5} = 6$ $3 y - \frac{x - 2}{7} = 9$

Step 1

Solve for $x$ in $3 x - \frac{y - 3}{5} = 6$ .

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Step 1.1

Simplify $3 x - \frac{y - 3}{5}$ .

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Step 1.1.1

To write $3 x$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

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

$3 y - \frac{x - 2}{7} = 9$

Step 1.1.2

Simplify terms.

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Step 1.1.2.1

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

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

$3 y - \frac{x - 2}{7} = 9$

Step 1.1.2.2

Combine the numerators over the common denominator.

$\frac{3 x \cdot 5 - (y - 3)}{5} = 6$

$3 y - \frac{x - 2}{7} = 9$

$\frac{3 x \cdot 5 - (y - 3)}{5} = 6$

$3 y - \frac{x - 2}{7} = 9$

Step 1.1.3

Simplify the numerator.

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Step 1.1.3.1

Multiply $5$ by $3$ .

$\frac{15 x - (y - 3)}{5} = 6$

$3 y - \frac{x - 2}{7} = 9$

Step 1.1.3.2

Apply the distributive property.

$\frac{15 x - y + 3}{5} = 6$

$3 y - \frac{x - 2}{7} = 9$

Step 1.1.3.3

Multiply $- 1$ by $- 3$ .

$\frac{15 x - y + 3}{5} = 6$

$3 y - \frac{x - 2}{7} = 9$

$\frac{15 x - y + 3}{5} = 6$

$3 y - \frac{x - 2}{7} = 9$

$\frac{15 x - y + 3}{5} = 6$

$3 y - \frac{x - 2}{7} = 9$

Step 1.2

Multiply both sides by $5$ .

$\frac{15 x - y + 3}{5} \cdot 5 = 6 \cdot 5$

$3 y - \frac{x - 2}{7} = 9$

Step 1.3

Simplify.

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Step 1.3.1

Simplify the left side.

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Step 1.3.1.1

Cancel the common factor of $5$ .

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Step 1.3.1.1.1

Cancel the common factor.

$\frac{15 x - y + 3}{5} \cdot 5 = 6 \cdot 5$

$3 y - \frac{x - 2}{7} = 9$

Step 1.3.1.1.2

Rewrite the expression.

$15 x - y + 3 = 6 \cdot 5$

$3 y - \frac{x - 2}{7} = 9$

$15 x - y + 3 = 6 \cdot 5$

$3 y - \frac{x - 2}{7} = 9$

$15 x - y + 3 = 6 \cdot 5$

$3 y - \frac{x - 2}{7} = 9$

Step 1.3.2

Simplify the right side.

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Step 1.3.2.1

Multiply $6$ by $5$ .

$15 x - y + 3 = 30$

$3 y - \frac{x - 2}{7} = 9$

$15 x - y + 3 = 30$

$3 y - \frac{x - 2}{7} = 9$

$15 x - y + 3 = 30$

$3 y - \frac{x - 2}{7} = 9$

Step 1.4

Solve for $x$ .

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Step 1.4.1

Move all terms not containing $x$ to the right side of the equation.

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Step 1.4.1.1

Add $y$ to both sides of the equation.

$15 x + 3 = 30 + y$

$3 y - \frac{x - 2}{7} = 9$

Step 1.4.1.2

Subtract $3$ from both sides of the equation.

$15 x = 30 + y - 3$

$3 y - \frac{x - 2}{7} = 9$

Step 1.4.1.3

Subtract $3$ from $30$ .

$15 x = y + 27$

$3 y - \frac{x - 2}{7} = 9$

$15 x = y + 27$

$3 y - \frac{x - 2}{7} = 9$

Step 1.4.2

Divide each term in $15 x = y + 27$ by $15$ and simplify.

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Step 1.4.2.1

Divide each term in $15 x = y + 27$ by $15$ .

$\frac{15 x}{15} = \frac{y}{15} + \frac{27}{15}$

$3 y - \frac{x - 2}{7} = 9$

Step 1.4.2.2

Simplify the left side.

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Step 1.4.2.2.1

Cancel the common factor of $15$ .

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Step 1.4.2.2.1.1

Cancel the common factor.

$\frac{15 x}{15} = \frac{y}{15} + \frac{27}{15}$

$3 y - \frac{x - 2}{7} = 9$

Step 1.4.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{y}{15} + \frac{27}{15}$

$3 y - \frac{x - 2}{7} = 9$

$x = \frac{y}{15} + \frac{27}{15}$

$3 y - \frac{x - 2}{7} = 9$

$x = \frac{y}{15} + \frac{27}{15}$

$3 y - \frac{x - 2}{7} = 9$

Step 1.4.2.3

Simplify the right side.

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Step 1.4.2.3.1

Cancel the common factor of $27$ and $15$ .

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Step 1.4.2.3.1.1

Factor $3$ out of $27$ .

$x = \frac{y}{15} + \frac{3 (9)}{15}$

$3 y - \frac{x - 2}{7} = 9$

Step 1.4.2.3.1.2

Cancel the common factors.

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Step 1.4.2.3.1.2.1

Factor $3$ out of $15$ .

$x = \frac{y}{15} + \frac{3 \cdot 9}{3 \cdot 5}$

$3 y - \frac{x - 2}{7} = 9$

Step 1.4.2.3.1.2.2

Cancel the common factor.

$x = \frac{y}{15} + \frac{3 \cdot 9}{3 \cdot 5}$

$3 y - \frac{x - 2}{7} = 9$

Step 1.4.2.3.1.2.3

Rewrite the expression.

$x = \frac{y}{15} + \frac{9}{5}$

$3 y - \frac{x - 2}{7} = 9$

$x = \frac{y}{15} + \frac{9}{5}$

$3 y - \frac{x - 2}{7} = 9$

$x = \frac{y}{15} + \frac{9}{5}$

$3 y - \frac{x - 2}{7} = 9$

$x = \frac{y}{15} + \frac{9}{5}$

$3 y - \frac{x - 2}{7} = 9$

$x = \frac{y}{15} + \frac{9}{5}$

$3 y - \frac{x - 2}{7} = 9$

$x = \frac{y}{15} + \frac{9}{5}$

$3 y - \frac{x - 2}{7} = 9$

$x = \frac{y}{15} + \frac{9}{5}$

$3 y - \frac{x - 2}{7} = 9$

Step 2

Replace all occurrences of $x$ with $\frac{y}{15} + \frac{9}{5}$ in each equation.

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Step 2.1

Replace all occurrences of $x$ in $3 y - \frac{x - 2}{7} = 9$ with $\frac{y}{15} + \frac{9}{5}$ .

$3 y - \frac{(\frac{y}{15} + \frac{9}{5}) - 2}{7} = 9$

$x = \frac{y}{15} + \frac{9}{5}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $3 y - \frac{(\frac{y}{15} + \frac{9}{5}) - 2}{7}$ .

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Step 2.2.1.1

Simplify each term.

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Step 2.2.1.1.1

Simplify the numerator.

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Step 2.2.1.1.1.1

To write $- 2$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$3 y - \frac{\frac{y}{15} + \frac{9}{5} - 2 \cdot \frac{5}{5}}{7} = 9$

$x = \frac{y}{15} + \frac{9}{5}$

Step 2.2.1.1.1.2

Combine $- 2$ and $\frac{5}{5}$ .

$3 y - \frac{\frac{y}{15} + \frac{9}{5} + \frac{- 2 \cdot 5}{5}}{7} = 9$

$x = \frac{y}{15} + \frac{9}{5}$

Step 2.2.1.1.1.3

Combine the numerators over the common denominator.

$3 y - \frac{\frac{y}{15} + \frac{9 - 2 \cdot 5}{5}}{7} = 9$

$x = \frac{y}{15} + \frac{9}{5}$

Step 2.2.1.1.1.4

Simplify the numerator.

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Step 2.2.1.1.1.4.1

Multiply $- 2$ by $5$ .

$3 y - \frac{\frac{y}{15} + \frac{9 - 10}{5}}{7} = 9$

$x = \frac{y}{15} + \frac{9}{5}$

Step 2.2.1.1.1.4.2

Subtract $10$ from $9$ .

$3 y - \frac{\frac{y}{15} + \frac{- 1}{5}}{7} = 9$

$x = \frac{y}{15} + \frac{9}{5}$

$3 y - \frac{\frac{y}{15} + \frac{- 1}{5}}{7} = 9$

$x = \frac{y}{15} + \frac{9}{5}$

Step 2.2.1.1.1.5

Move the negative in front of the fraction.

$3 y - \frac{\frac{y}{15} - \frac{1}{5}}{7} = 9$

$x = \frac{y}{15} + \frac{9}{5}$

Step 2.2.1.1.1.6

To write $- \frac{1}{5}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$3 y - \frac{\frac{y}{15} - \frac{1}{5} \cdot \frac{3}{3}}{7} = 9$

$x = \frac{y}{15} + \frac{9}{5}$

Step 2.2.1.1.1.7

Write each expression with a common denominator of $15$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.2.1.1.1.7.1

Multiply $\frac{1}{5}$ by $\frac{3}{3}$ .

$3 y - \frac{\frac{y}{15} - \frac{3}{5 \cdot 3}}{7} = 9$

$x = \frac{y}{15} + \frac{9}{5}$

Step 2.2.1.1.1.7.2

Multiply $5$ by $3$ .

$3 y - \frac{\frac{y}{15} - \frac{3}{15}}{7} = 9$

$x = \frac{y}{15} + \frac{9}{5}$

$3 y - \frac{\frac{y}{15} - \frac{3}{15}}{7} = 9$

$x = \frac{y}{15} + \frac{9}{5}$

Step 2.2.1.1.1.8

Combine the numerators over the common denominator.

$3 y - \frac{\frac{y - 3}{15}}{7} = 9$

$x = \frac{y}{15} + \frac{9}{5}$

$3 y - \frac{\frac{y - 3}{15}}{7} = 9$

$x = \frac{y}{15} + \frac{9}{5}$

Step 2.2.1.1.2

Multiply the numerator by the reciprocal of the denominator.

$3 y - (\frac{y - 3}{15} \cdot \frac{1}{7}) = 9$

$x = \frac{y}{15} + \frac{9}{5}$

Step 2.2.1.1.3

Multiply $\frac{y - 3}{15} \cdot \frac{1}{7}$ .

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Step 2.2.1.1.3.1

Multiply $\frac{y - 3}{15}$ by $\frac{1}{7}$ .

$3 y - \frac{y - 3}{15 \cdot 7} = 9$

$x = \frac{y}{15} + \frac{9}{5}$

Step 2.2.1.1.3.2

Multiply $15$ by $7$ .

$3 y - \frac{y - 3}{105} = 9$

$x = \frac{y}{15} + \frac{9}{5}$

$3 y - \frac{y - 3}{105} = 9$

$x = \frac{y}{15} + \frac{9}{5}$

$3 y - \frac{y - 3}{105} = 9$

$x = \frac{y}{15} + \frac{9}{5}$

Step 2.2.1.2

To write $3 y$ as a fraction with a common denominator, multiply by $\frac{105}{105}$ .

$3 y \cdot \frac{105}{105} - \frac{y - 3}{105} = 9$

$x = \frac{y}{15} + \frac{9}{5}$

Step 2.2.1.3

Simplify terms.

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Step 2.2.1.3.1

Combine $3 y$ and $\frac{105}{105}$ .

$\frac{3 y \cdot 105}{105} - \frac{y - 3}{105} = 9$

$x = \frac{y}{15} + \frac{9}{5}$

Step 2.2.1.3.2

Combine the numerators over the common denominator.

$\frac{3 y \cdot 105 - (y - 3)}{105} = 9$

$x = \frac{y}{15} + \frac{9}{5}$

$\frac{3 y \cdot 105 - (y - 3)}{105} = 9$

$x = \frac{y}{15} + \frac{9}{5}$

Step 2.2.1.4

Simplify the numerator.

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Step 2.2.1.4.1

Multiply $105$ by $3$ .

$\frac{315 y - (y - 3)}{105} = 9$

$x = \frac{y}{15} + \frac{9}{5}$

Step 2.2.1.4.2

Apply the distributive property.

$\frac{315 y - y + 3}{105} = 9$

$x = \frac{y}{15} + \frac{9}{5}$

Step 2.2.1.4.3

Multiply $- 1$ by $- 3$ .

$\frac{315 y - y + 3}{105} = 9$

$x = \frac{y}{15} + \frac{9}{5}$

Step 2.2.1.4.4

Subtract $y$ from $315 y$ .

$\frac{314 y + 3}{105} = 9$

$x = \frac{y}{15} + \frac{9}{5}$

$\frac{314 y + 3}{105} = 9$

$x = \frac{y}{15} + \frac{9}{5}$

$\frac{314 y + 3}{105} = 9$

$x = \frac{y}{15} + \frac{9}{5}$

$\frac{314 y + 3}{105} = 9$

$x = \frac{y}{15} + \frac{9}{5}$

$\frac{314 y + 3}{105} = 9$

$x = \frac{y}{15} + \frac{9}{5}$

Step 3

Solve for $y$ in $\frac{314 y + 3}{105} = 9$ .

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Step 3.1

Multiply both sides by $105$ .

$\frac{314 y + 3}{105} \cdot 105 = 9 \cdot 105$

$x = \frac{y}{15} + \frac{9}{5}$

Step 3.2

Simplify.

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Step 3.2.1

Simplify the left side.

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Step 3.2.1.1

Cancel the common factor of $105$ .

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Step 3.2.1.1.1

Cancel the common factor.

$\frac{314 y + 3}{105} \cdot 105 = 9 \cdot 105$

$x = \frac{y}{15} + \frac{9}{5}$

Step 3.2.1.1.2

Rewrite the expression.

$314 y + 3 = 9 \cdot 105$

$x = \frac{y}{15} + \frac{9}{5}$

$314 y + 3 = 9 \cdot 105$

$x = \frac{y}{15} + \frac{9}{5}$

$314 y + 3 = 9 \cdot 105$

$x = \frac{y}{15} + \frac{9}{5}$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Multiply $9$ by $105$ .

$314 y + 3 = 945$

$x = \frac{y}{15} + \frac{9}{5}$

$314 y + 3 = 945$

$x = \frac{y}{15} + \frac{9}{5}$

$314 y + 3 = 945$

$x = \frac{y}{15} + \frac{9}{5}$

Step 3.3

Solve for $y$ .

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Step 3.3.1

Move all terms not containing $y$ to the right side of the equation.

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Step 3.3.1.1

Subtract $3$ from both sides of the equation.

$314 y = 945 - 3$

$x = \frac{y}{15} + \frac{9}{5}$

Step 3.3.1.2

Subtract $3$ from $945$ .

$314 y = 942$

$x = \frac{y}{15} + \frac{9}{5}$

$314 y = 942$

$x = \frac{y}{15} + \frac{9}{5}$

Step 3.3.2

Divide each term in $314 y = 942$ by $314$ and simplify.

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Step 3.3.2.1

Divide each term in $314 y = 942$ by $314$ .

$\frac{314 y}{314} = \frac{942}{314}$

$x = \frac{y}{15} + \frac{9}{5}$

Step 3.3.2.2

Simplify the left side.

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Step 3.3.2.2.1

Cancel the common factor of $314$ .

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Step 3.3.2.2.1.1

Cancel the common factor.

$\frac{314 y}{314} = \frac{942}{314}$

$x = \frac{y}{15} + \frac{9}{5}$

Step 3.3.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{942}{314}$

$x = \frac{y}{15} + \frac{9}{5}$

$y = \frac{942}{314}$

$x = \frac{y}{15} + \frac{9}{5}$

$y = \frac{942}{314}$

$x = \frac{y}{15} + \frac{9}{5}$

Step 3.3.2.3

Simplify the right side.

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Step 3.3.2.3.1

Divide $942$ by $314$ .

$y = 3$

$x = \frac{y}{15} + \frac{9}{5}$

$y = 3$

$x = \frac{y}{15} + \frac{9}{5}$

$y = 3$

$x = \frac{y}{15} + \frac{9}{5}$

$y = 3$

$x = \frac{y}{15} + \frac{9}{5}$

$y = 3$

$x = \frac{y}{15} + \frac{9}{5}$

Step 4

Replace all occurrences of $y$ with $3$ in each equation.

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Step 4.1

Replace all occurrences of $y$ in $x = \frac{y}{15} + \frac{9}{5}$ with $3$ .

$x = \frac{3}{15} + \frac{9}{5}$

$y = 3$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $\frac{3}{15} + \frac{9}{5}$ .

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Step 4.2.1.1

Cancel the common factor of $3$ and $15$ .

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Step 4.2.1.1.1

Factor $3$ out of $3$ .

$x = \frac{3 (1)}{15} + \frac{9}{5}$

$y = 3$

Step 4.2.1.1.2

Cancel the common factors.

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Step 4.2.1.1.2.1

Factor $3$ out of $15$ .

$x = \frac{3 \cdot 1}{3 \cdot 5} + \frac{9}{5}$

$y = 3$

Step 4.2.1.1.2.2

Cancel the common factor.

$x = \frac{3 \cdot 1}{3 \cdot 5} + \frac{9}{5}$

$y = 3$

Step 4.2.1.1.2.3

Rewrite the expression.

$x = \frac{1}{5} + \frac{9}{5}$

$y = 3$

$x = \frac{1}{5} + \frac{9}{5}$

$y = 3$

$x = \frac{1}{5} + \frac{9}{5}$

$y = 3$

Step 4.2.1.2

Combine the numerators over the common denominator.

$x = \frac{1 + 9}{5}$

$y = 3$

Step 4.2.1.3

Simplify the expression.

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Step 4.2.1.3.1

Add $1$ and $9$ .

$x = \frac{10}{5}$

$y = 3$

Step 4.2.1.3.2

Divide $10$ by $5$ .

$x = 2$

$y = 3$

$x = 2$

$y = 3$

$x = 2$

$y = 3$

$x = 2$

$y = 3$

$x = 2$

$y = 3$

Step 5

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

$(2, 3)$

Step 6

The result can be shown in multiple forms.

Point Form:

$(2, 3)$

Equation Form:

$x = 2, y = 3$

Step 7