Pre-Algebra Examples

$\frac{5 x}{2} + \frac{5 y}{3} = \frac{65}{5}$

Step 1

Solve the equation for $y$ .

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Step 1.1

Divide $65$ by $5$ .

$\frac{5 x}{2} + \frac{5 y}{3} = 13$

Step 1.2

Subtract $\frac{5 x}{2}$ from both sides of the equation.

$\frac{5 y}{3} = 13 - \frac{5 x}{2}$

Step 1.3

Multiply both sides of the equation by $\frac{3}{5}$ .

$\frac{3}{5} \cdot \frac{5 y}{3} = \frac{3}{5} (13 - \frac{5 x}{2})$

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

Simplify $\frac{3}{5} \cdot \frac{5 y}{3}$ .

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Step 1.4.1.1.1

Cancel the common factor of $3$ .

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Step 1.4.1.1.1.1

Cancel the common factor.

$\frac{3}{5} \cdot \frac{5 y}{3} = \frac{3}{5} (13 - \frac{5 x}{2})$

Step 1.4.1.1.1.2

Rewrite the expression.

$\frac{1}{5} (5 y) = \frac{3}{5} (13 - \frac{5 x}{2})$

Step 1.4.1.1.2

Cancel the common factor of $5$ .

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Step 1.4.1.1.2.1

Factor $5$ out of $5 y$ .

$\frac{1}{5} (5 (y)) = \frac{3}{5} (13 - \frac{5 x}{2})$

Step 1.4.1.1.2.2

Cancel the common factor.

$\frac{1}{5} (5 y) = \frac{3}{5} (13 - \frac{5 x}{2})$

Step 1.4.1.1.2.3

Rewrite the expression.

$y = \frac{3}{5} (13 - \frac{5 x}{2})$

Step 1.4.2

Simplify the right side.

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Step 1.4.2.1

Simplify $\frac{3}{5} (13 - \frac{5 x}{2})$ .

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Step 1.4.2.1.1

Apply the distributive property.

$y = \frac{3}{5} \cdot 13 + \frac{3}{5} (- \frac{5 x}{2})$

Step 1.4.2.1.2

Multiply $\frac{3}{5} \cdot 13$ .

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Step 1.4.2.1.2.1

Combine $\frac{3}{5}$ and $13$ .

$y = \frac{3 \cdot 13}{5} + \frac{3}{5} (- \frac{5 x}{2})$

Step 1.4.2.1.2.2

Multiply $3$ by $13$ .

$y = \frac{39}{5} + \frac{3}{5} (- \frac{5 x}{2})$

Step 1.4.2.1.3

Cancel the common factor of $5$ .

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Step 1.4.2.1.3.1

Move the leading negative in $- \frac{5 x}{2}$ into the numerator.

$y = \frac{39}{5} + \frac{3}{5} \cdot \frac{- 5 x}{2}$

Step 1.4.2.1.3.2

Factor $5$ out of $- 5 x$ .

$y = \frac{39}{5} + \frac{3}{5} \cdot \frac{5 (- x)}{2}$

Step 1.4.2.1.3.3

Cancel the common factor.

$y = \frac{39}{5} + \frac{3}{5} \cdot \frac{5 (- x)}{2}$

Step 1.4.2.1.3.4

Rewrite the expression.

$y = \frac{39}{5} + 3 \frac{- x}{2}$

Step 1.4.2.1.4

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

$y = \frac{39}{5} + \frac{3 (- x)}{2}$

Step 1.4.2.1.5

Simplify the expression.

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Step 1.4.2.1.5.1

Multiply $- 1$ by $3$ .

$y = \frac{39}{5} + \frac{- 3 x}{2}$

Step 1.4.2.1.5.2

Move the negative in front of the fraction.

$y = \frac{39}{5} - \frac{3 x}{2}$

Step 1.5

Reorder $\frac{39}{5}$ and $- \frac{3 x}{2}$ .

$y = - \frac{3 x}{2} + \frac{39}{5}$

Step 2

Choose any value for $x$ that is in the domain to plug into the equation.

Step 3

Choose $0$ to substitute in for $x$ to find the ordered pair.

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Step 3.1

Remove parentheses.

$y = - \frac{3 (0)}{2} + \frac{39}{5}$

Step 3.2

Simplify $- \frac{3 (0)}{2} + \frac{39}{5}$ .

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Step 3.2.1

Simplify each term.

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Step 3.2.1.1

Cancel the common factor of $0$ and $2$ .

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Step 3.2.1.1.1

Factor $2$ out of $3 (0)$ .

$y = - \frac{2 (3 \cdot (0))}{2} + \frac{39}{5}$

Step 3.2.1.1.2

Cancel the common factors.

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Step 3.2.1.1.2.1

Factor $2$ out of $2$ .

$y = - \frac{2 (3 \cdot (0))}{2 (1)} + \frac{39}{5}$

Step 3.2.1.1.2.2

Cancel the common factor.

$y = - \frac{2 (3 \cdot (0))}{2 \cdot 1} + \frac{39}{5}$

Step 3.2.1.1.2.3

Rewrite the expression.

$y = - \frac{3 \cdot (0)}{1} + \frac{39}{5}$

Step 3.2.1.1.2.4

Divide $3 \cdot (0)$ by $1$ .

$y = - (3 \cdot (0)) + \frac{39}{5}$

Step 3.2.1.2

Multiply $- (3 \cdot (0))$ .

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Step 3.2.1.2.1

Multiply $3$ by $0$ .

$y = - 0 + \frac{39}{5}$

Step 3.2.1.2.2

Multiply $- 1$ by $0$ .

$y = 0 + \frac{39}{5}$

Step 3.2.2

Add $0$ and $\frac{39}{5}$ .

$y = \frac{39}{5}$

Step 3.3

Use the $x$ and $y$ values to form the ordered pair.

$(0, \frac{39}{5})$

Step 4

Choose $1$ to substitute in for $x$ to find the ordered pair.

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Step 4.1

Remove parentheses.

$y = - \frac{3 (1)}{2} + \frac{39}{5}$

Step 4.2

Simplify $- \frac{3 (1)}{2} + \frac{39}{5}$ .

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Step 4.2.1

Multiply $3$ by $1$ .

$y = - \frac{3}{2} + \frac{39}{5}$

Step 4.2.2

To write $- \frac{3}{2}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$y = - \frac{3}{2} \cdot \frac{5}{5} + \frac{39}{5}$

Step 4.2.3

To write $\frac{39}{5}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y = - \frac{3}{2} \cdot \frac{5}{5} + \frac{39}{5} \cdot \frac{2}{2}$

Step 4.2.4

Write each expression with a common denominator of $10$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.2.4.1

Multiply $\frac{3}{2}$ by $\frac{5}{5}$ .

$y = - \frac{3 \cdot 5}{2 \cdot 5} + \frac{39}{5} \cdot \frac{2}{2}$

Step 4.2.4.2

Multiply $2$ by $5$ .

$y = - \frac{3 \cdot 5}{10} + \frac{39}{5} \cdot \frac{2}{2}$

Step 4.2.4.3

Multiply $\frac{39}{5}$ by $\frac{2}{2}$ .

$y = - \frac{3 \cdot 5}{10} + \frac{39 \cdot 2}{5 \cdot 2}$

Step 4.2.4.4

Multiply $5$ by $2$ .

$y = - \frac{3 \cdot 5}{10} + \frac{39 \cdot 2}{10}$

Step 4.2.5

Combine the numerators over the common denominator.

$y = \frac{- 3 \cdot 5 + 39 \cdot 2}{10}$

Step 4.2.6

Simplify the numerator.

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Step 4.2.6.1

Multiply $- 3$ by $5$ .

$y = \frac{- 15 + 39 \cdot 2}{10}$

Step 4.2.6.2

Multiply $39$ by $2$ .

$y = \frac{- 15 + 78}{10}$

Step 4.2.6.3

Add $- 15$ and $78$ .

$y = \frac{63}{10}$

Step 4.3

Use the $x$ and $y$ values to form the ordered pair.

$(1, \frac{63}{10})$

Step 5

Choose $2$ to substitute in for $x$ to find the ordered pair.

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Step 5.1

Remove parentheses.

$y = - \frac{3 (2)}{2} + \frac{39}{5}$

Step 5.2

Simplify $- \frac{3 (2)}{2} + \frac{39}{5}$ .

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Step 5.2.1

Simplify each term.

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Step 5.2.1.1

Cancel the common factor of $2$ .

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Step 5.2.1.1.1

Cancel the common factor.

$y = - \frac{3 \cdot 2}{2} + \frac{39}{5}$

Step 5.2.1.1.2

Divide $3$ by $1$ .

$y = - 1 \cdot 3 + \frac{39}{5}$

Step 5.2.1.2

Multiply $- 1$ by $3$ .

$y = - 3 + \frac{39}{5}$

Step 5.2.2

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

$y = - 3 \cdot \frac{5}{5} + \frac{39}{5}$

Step 5.2.3

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

$y = \frac{- 3 \cdot 5}{5} + \frac{39}{5}$

Step 5.2.4

Combine the numerators over the common denominator.

$y = \frac{- 3 \cdot 5 + 39}{5}$

Step 5.2.5

Simplify the numerator.

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Step 5.2.5.1

Multiply $- 3$ by $5$ .

$y = \frac{- 15 + 39}{5}$

Step 5.2.5.2

Add $- 15$ and $39$ .

$y = \frac{24}{5}$

Step 5.3

Use the $x$ and $y$ values to form the ordered pair.

$(2, \frac{24}{5})$

Step 6

These are three possible solutions to the equation.

$(0, \frac{39}{5}), (1, \frac{63}{10}), (2, \frac{24}{5})$

Step 7