Precalculus Examples

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

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

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

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

Simplify the right side of the equation.

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To write $\frac{5}{1}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Write each expression with a common denominator of $2$ , by multiplying each by an appropriate factor of $1$ .

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Combine.

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

Multiply $2$ by $1$ .

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

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Multiply $- 1$ by $5$ .

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

Multiply $5$ by $2$ .

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

Add $- 5$ and $10$ .

$\frac{5}{3 y} = \frac{5}{2}$

Set up the rational expression with the same denominator over the entire equation.

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Multiply each term by a factor of $1$ that will equate all the denominators. In this case, all terms need a denominator of $6 y$ . The $\frac{5}{3 y}$ expression needs to be multiplied by $\frac{2}{2}$ to make the denominator $6 y$ . The $\frac{5}{2}$ expression needs to be multiplied by $\frac{3 y}{3 y}$ to make the denominator $6 y$ .

$\frac{5}{3 y} \cdot \frac{2}{2} = \frac{5}{2} \cdot \frac{3 y}{3 y}$

Multiply the expression by a factor of $1$ to create the least common denominator (LCD) of $6 y$ .

$5 (2)$

Multiply $5$ by $2$ .

$\frac{10}{6 y} = \frac{5}{2} \cdot \frac{3 y}{3 y}$

Multiply the expression by a factor of $1$ to create the least common denominator (LCD) of $6 y$ .

$5 (3 y)$

Multiply $3$ by $5$ .

$\frac{10}{6 y} = \frac{15 y}{6 y}$

Since the expression on each side of the equation has the same denominator, the numerators must be equal.

$10 = 15 y$

Rewrite the equation as $15 y = 10$ .

$15 y = 10$

Divide each term by $15$ and simplify.

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Divide each term in $15 y = 10$ by $15$ .

$\frac{15 y}{15} = \frac{10}{15}$

Reduce the expression by cancelling the common factors.

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Cancel the common factor.

$\frac{15 y}{15} = \frac{10}{15}$

Divide $y$ by $1$ .

$y = \frac{10}{15}$

Reduce the expression by cancelling the common factors.

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Factor $5$ out of $15$ .

$y = \frac{5 \cdot 2}{5 \cdot 3}$

Cancel the common factor.

$y = \frac{5 \cdot 2}{5 \cdot 3}$

Rewrite the expression.

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

Verify each of the solutions by substituting them into $\frac{5}{3 y} + \frac{5}{2} = 5$ and solving.

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

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