Precalculus Examples

$- \frac{4}{5} - \frac{2}{4 y} = 1$

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

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Add $\frac{4}{5}$ to both sides of the equation.

$- \frac{2}{4 y} = \frac{4}{5} + 1$

Simplify the right side of the equation.

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

$- \frac{2}{4 y} = \frac{4}{5} + \frac{1}{1} \cdot \frac{5}{5}$

Write each expression with a common denominator of $5$ , by multiplying each by an appropriate factor of $1$ .

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Combine.

$- \frac{2}{4 y} = \frac{4}{5} + \frac{1 \cdot 5}{1 \cdot 5}$

Multiply $5$ by $1$ .

$- \frac{2}{4 y} = \frac{4}{5} + \frac{1 \cdot 5}{5}$

Combine the numerators over the common denominator.

$- \frac{2}{4 y} = \frac{4 + 1 \cdot 5}{5}$

Simplify the numerator.

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Multiply $1$ by $5$ .

$- \frac{2}{4 y} = \frac{4 + 5}{5}$

Add $4$ and $5$ .

$- \frac{2}{4 y} = \frac{9}{5}$

Reduce the expression by cancelling the common factors.

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Factor $2$ out of $4 y$ .

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Factor $2$ out of $4$ .

$- \frac{2 \cdot 1}{2 \cdot (2 y)} = \frac{9}{5}$

Factor $2$ out of $4$ .

$- \frac{2 (1)}{2 (2) y} = \frac{9}{5}$

Move $2$ .

$- \frac{2 (1)}{2 ((2) y)} = \frac{9}{5}$

Multiply $2$ by $y$ .

$- \frac{2 (1)}{2 (2 y)} = \frac{9}{5}$

Cancel the common factor.

$- \frac{2 \cdot 1}{2 (2 y)} = \frac{9}{5}$

Rewrite the expression.

$- \frac{1}{2 y} = \frac{9}{5}$

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 $10 y$ . The $- \frac{1}{2 y}$ expression needs to be multiplied by $\frac{5}{5}$ to make the denominator $10 y$ . The $\frac{9}{5}$ expression needs to be multiplied by $\frac{2 y}{2 y}$ to make the denominator $10 y$ .

$- \frac{1}{2 y} \cdot \frac{5}{5} = \frac{9}{5} \cdot \frac{2 y}{2 y}$

Multiply the expression by a factor of $1$ to create the least common denominator (LCD) of $10 y$ .

$1 (5)$

Multiply $5$ by $1$ .

$- \frac{5}{10 y} = \frac{9}{5} \cdot \frac{2 y}{2 y}$

Multiply the expression by a factor of $1$ to create the least common denominator (LCD) of $10 y$ .

$9 (2 y)$

Multiply $2$ by $9$ .

$- \frac{5}{10 y} = \frac{18 y}{10 y}$

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

$- (5) = 18 y$

Rewrite the equation as $18 y = - (5)$ .

$18 y = - (5)$

Multiply $- 1$ by $5$ .

$18 y = - 5$

Divide each term by $18$ and simplify.

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Divide each term in $18 y = - 5$ by $18$ .

$\frac{18 y}{18} = - \frac{5}{18}$

Reduce the expression by cancelling the common factors.

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Cancel the common factor.

$\frac{18 y}{18} = - \frac{5}{18}$

Divide $y$ by $1$ .

$y = - \frac{5}{18}$

Verify each of the solutions by substituting them into $- \frac{4}{5} - \frac{2}{4 y} = 1$ and solving.

$y = - \frac{5}{18}$

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