Basic Math Examples

$15 \div (19 + y) + 15 \div (19 - y) = 1.5$

Step 1

Simplify $15 \div (19 + y) + 15 \div (19 - y)$ .

Tap for more steps...

Step 1.1

Simplify each term.

Tap for more steps...

Step 1.1.1

Rewrite the division as a fraction.

$\frac{15}{19 + y} + 15 \div (19 - y) = 1.5$

Step 1.1.2

Rewrite the division as a fraction.

$\frac{15}{19 + y} + \frac{15}{19 - y} = 1.5$

Step 1.2

To write $\frac{15}{19 + y}$ as a fraction with a common denominator, multiply by $\frac{19 - y}{19 - y}$ .

$\frac{15}{19 + y} \cdot \frac{19 - y}{19 - y} + \frac{15}{19 - y} = 1.5$

Step 1.3

To write $\frac{15}{19 - y}$ as a fraction with a common denominator, multiply by $\frac{19 + y}{19 + y}$ .

$\frac{15}{19 + y} \cdot \frac{19 - y}{19 - y} + \frac{15}{19 - y} \cdot \frac{19 + y}{19 + y} = 1.5$

Step 1.4

Write each expression with a common denominator of $(19 + y) (19 - y)$ , by multiplying each by an appropriate factor of $1$ .

Tap for more steps...

Step 1.4.1

Multiply $\frac{15}{19 + y}$ by $\frac{19 - y}{19 - y}$ .

$\frac{15 (19 - y)}{(19 + y) (19 - y)} + \frac{15}{19 - y} \cdot \frac{19 + y}{19 + y} = 1.5$

Step 1.4.2

Multiply $\frac{15}{19 - y}$ by $\frac{19 + y}{19 + y}$ .

$\frac{15 (19 - y)}{(19 + y) (19 - y)} + \frac{15 (19 + y)}{(19 - y) (19 + y)} = 1.5$

Step 1.4.3

Reorder the factors of $(19 - y) (19 + y)$ .

$\frac{15 (19 - y)}{(19 + y) (19 - y)} + \frac{15 (19 + y)}{(19 + y) (19 - y)} = 1.5$

Step 1.5

Combine the numerators over the common denominator.

$\frac{15 (19 - y) + 15 (19 + y)}{(19 + y) (19 - y)} = 1.5$

Step 1.6

Simplify the numerator.

Tap for more steps...

Step 1.6.1

Factor $15$ out of $15 (19 - y) + 15 (19 + y)$ .

$\frac{15 (19 - y + 19 + y)}{(19 + y) (19 - y)} = 1.5$

Step 1.6.2

Add $19$ and $19$ .

$\frac{15 (- y + 38 + y)}{(19 + y) (19 - y)} = 1.5$

Step 1.6.3

Add $- y$ and $y$ .

$\frac{15 (0 + 38)}{(19 + y) (19 - y)} = 1.5$

Step 1.6.4

Add $0$ and $38$ .

$\frac{15 \cdot 38}{(19 + y) (19 - y)} = 1.5$

Step 1.7

Multiply $15$ by $38$ .

$\frac{570}{(19 + y) (19 - y)} = 1.5$

Step 2

Find the LCD of the terms in the equation.

Tap for more steps...

Step 2.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$(19 + y) (19 - y), 1$

Step 2.2

The LCM of one and any expression is the expression.

$(19 + y) (19 - y)$

Step 3

Multiply each term in $\frac{570}{(19 + y) (19 - y)} = 1.5$ by $(19 + y) (19 - y)$ to eliminate the fractions.

Tap for more steps...

Step 3.1

Multiply each term in $\frac{570}{(19 + y) (19 - y)} = 1.5$ by $(19 + y) (19 - y)$ .

$\frac{570}{(19 + y) (19 - y)} ((19 + y) (19 - y)) = 1.5 ((19 + y) (19 - y))$

Step 3.2

Simplify the left side.

Tap for more steps...

Step 3.2.1

Cancel the common factor of $(19 + y) (19 - y)$ .

Tap for more steps...

Step 3.2.1.1

Cancel the common factor.

$\frac{570}{(19 + y) (19 - y)} ((19 + y) (19 - y)) = 1.5 ((19 + y) (19 - y))$

Step 3.2.1.2

Rewrite the expression.

$570 = 1.5 ((19 + y) (19 - y))$

Step 3.3

Simplify the right side.

Tap for more steps...

Step 3.3.1

Expand $(19 + y) (19 - y)$ using the FOIL Method.

Tap for more steps...

Step 3.3.1.1

Apply the distributive property.

$570 = 1.5 (19 (19 - y) + y (19 - y))$

Step 3.3.1.2

Apply the distributive property.

$570 = 1.5 (19 \cdot 19 + 19 (- y) + y (19 - y))$

Step 3.3.1.3

Apply the distributive property.

$570 = 1.5 (19 \cdot 19 + 19 (- y) + y \cdot 19 + y (- y))$

Step 3.3.2

Simplify and combine like terms.

Tap for more steps...

Step 3.3.2.1

Simplify each term.

Tap for more steps...

Step 3.3.2.1.1

Multiply $19$ by $19$ .

$570 = 1.5 (361 + 19 (- y) + y \cdot 19 + y (- y))$

Step 3.3.2.1.2

Multiply $- 1$ by $19$ .

$570 = 1.5 (361 - 19 y + y \cdot 19 + y (- y))$

Step 3.3.2.1.3

Move $19$ to the left of $y$ .

$570 = 1.5 (361 - 19 y + 19 \cdot y + y (- y))$

Step 3.3.2.1.4

Rewrite using the commutative property of multiplication.

$570 = 1.5 (361 - 19 y + 19 y - y \cdot y)$

Step 3.3.2.1.5

Multiply $y$ by $y$ by adding the exponents.

Tap for more steps...

Step 3.3.2.1.5.1

Move $y$ .

$570 = 1.5 (361 - 19 y + 19 y - (y \cdot y))$

Step 3.3.2.1.5.2

Multiply $y$ by $y$ .

$570 = 1.5 (361 - 19 y + 19 y - y^{2})$

Step 3.3.2.2

Add $- 19 y$ and $19 y$ .

$570 = 1.5 (361 + 0 - y^{2})$

Step 3.3.2.3

Add $361$ and $0$ .

$570 = 1.5 (361 - y^{2})$

Step 3.3.3

Apply the distributive property.

$570 = 1.5 \cdot 361 + 1.5 (- y^{2})$

Step 3.3.4

Multiply.

Tap for more steps...

Step 3.3.4.1

Multiply $1.5$ by $361$ .

$570 = 541.5 + 1.5 (- y^{2})$

Step 3.3.4.2

Multiply $- 1$ by $1.5$ .

$570 = 541.5 - 1.5 y^{2}$

Step 4

Solve the equation.

Tap for more steps...

Step 4.1

Rewrite the equation as $541.5 - 1.5 y^{2} = 570$ .

$541.5 - 1.5 y^{2} = 570$

Step 4.2

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

Tap for more steps...

Step 4.2.1

Subtract $541.5$ from both sides of the equation.

$- 1.5 y^{2} = 570 - 541.5$

Step 4.2.2

Subtract $541.5$ from $570$ .

$- 1.5 y^{2} = 28.5$

Step 4.3

Divide each term in $- 1.5 y^{2} = 28.5$ by $- 1.5$ and simplify.

Tap for more steps...

Step 4.3.1

Divide each term in $- 1.5 y^{2} = 28.5$ by $- 1.5$ .

$\frac{- 1.5 y^{2}}{- 1.5} = \frac{28.5}{- 1.5}$

Step 4.3.2

Simplify the left side.

Tap for more steps...

Step 4.3.2.1

Cancel the common factor of $- 1.5$ .

Tap for more steps...

Step 4.3.2.1.1

Cancel the common factor.

$\frac{- 1.5 y^{2}}{- 1.5} = \frac{28.5}{- 1.5}$

Step 4.3.2.1.2

Divide $y^{2}$ by $1$ .

$y^{2} = \frac{28.5}{- 1.5}$

Step 4.3.3

Simplify the right side.

Tap for more steps...

Step 4.3.3.1

Divide $28.5$ by $- 1.5$ .

$y^{2} = - 19$

Step 4.4

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$y = \pm \sqrt{- 19}$

Step 4.5

Simplify $\pm \sqrt{- 19}$ .

Tap for more steps...

Step 4.5.1

Rewrite $- 19$ as $- 1 (19)$ .

$y = \pm \sqrt{- 1 (19)}$

Step 4.5.2

Rewrite $\sqrt{- 1 (19)}$ as $\sqrt{- 1} \cdot \sqrt{19}$ .

$y = \pm \sqrt{- 1} \cdot \sqrt{19}$

Step 4.5.3

Rewrite $\sqrt{- 1}$ as $i$ .

$y = \pm i \sqrt{19}$

Step 4.6

The complete solution is the result of both the positive and negative portions of the solution.

Tap for more steps...

Step 4.6.1

First, use the positive value of the $\pm$ to find the first solution.

$y = i \sqrt{19}$

Step 4.6.2

Next, use the negative value of the $\pm$ to find the second solution.

$y = - i \sqrt{19}$

Step 4.6.3

The complete solution is the result of both the positive and negative portions of the solution.

$y = i \sqrt{19}, - i \sqrt{19}$