Algebra Examples

$\frac{17 + y}{y} + \frac{32 + y}{y} = 100$

Step 1

Subtract $100$ from both sides of the equation.

$\frac{17 + y}{y} + \frac{32 + y}{y} - 100 = 0$

Step 2

Simplify $\frac{17 + y}{y} + \frac{32 + y}{y} - 100$ .

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Step 2.1

Combine the numerators over the common denominator.

$- 100 + \frac{17 + y + 32 + y}{y} = 0$

Step 2.2

Add $17$ and $32$ .

$- 100 + \frac{y + 49 + y}{y} = 0$

Step 2.3

Add $y$ and $y$ .

$- 100 + \frac{2 y + 49}{y} = 0$

Step 2.4

To write $- 100$ as a fraction with a common denominator, multiply by $\frac{y}{y}$ .

$- 100 \cdot \frac{y}{y} + \frac{2 y + 49}{y} = 0$

Step 2.5

Combine $- 100$ and $\frac{y}{y}$ .

$\frac{- 100 y}{y} + \frac{2 y + 49}{y} = 0$

Step 2.6

Combine the numerators over the common denominator.

$\frac{- 100 y + 2 y + 49}{y} = 0$

Step 2.7

Simplify the numerator.

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Step 2.7.1

Add $- 100 y$ and $2 y$ .

$\frac{- 98 y + 49}{y} = 0$

Step 2.7.2

Factor $49$ out of $- 98 y + 49$ .

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Step 2.7.2.1

Factor $49$ out of $- 98 y$ .

$\frac{49 (- 2 y) + 49}{y} = 0$

Step 2.7.2.2

Factor $49$ out of $49$ .

$\frac{49 (- 2 y) + 49 (1)}{y} = 0$

Step 2.7.2.3

Factor $49$ out of $49 (- 2 y) + 49 (1)$ .

$\frac{49 (- 2 y + 1)}{y} = 0$

Step 2.8

Factor $- 1$ out of $- 2 y$ .

$\frac{49 (- (2 y) + 1)}{y} = 0$

Step 2.9

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

$\frac{49 (- (2 y) - 1 (- 1))}{y} = 0$

Step 2.10

Factor $- 1$ out of $- (2 y) - 1 (- 1)$ .

$\frac{49 (- (2 y - 1))}{y} = 0$

Step 2.11

Rewrite $- (2 y - 1)$ as $- 1 (2 y - 1)$ .

$\frac{49 (- 1 (2 y - 1))}{y} = 0$

Step 2.12

Move the negative in front of the fraction.

$- \frac{49 (2 y - 1)}{y} = 0$

Step 3

Set the numerator equal to zero.

$49 (2 y - 1) = 0$

Step 4

Solve the equation for $y$ .

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Step 4.1

Divide each term in $49 (2 y - 1) = 0$ by $49$ and simplify.

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Step 4.1.1

Divide each term in $49 (2 y - 1) = 0$ by $49$ .

$\frac{49 (2 y - 1)}{49} = \frac{0}{49}$

Step 4.1.2

Simplify the left side.

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Step 4.1.2.1

Cancel the common factor of $49$ .

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Step 4.1.2.1.1

Cancel the common factor.

$\frac{49 (2 y - 1)}{49} = \frac{0}{49}$

Step 4.1.2.1.2

Divide $2 y - 1$ by $1$ .

$2 y - 1 = \frac{0}{49}$

Step 4.1.3

Simplify the right side.

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Step 4.1.3.1

Divide $0$ by $49$ .

$2 y - 1 = 0$

Step 4.2

Add $1$ to both sides of the equation.

$2 y = 1$

Step 4.3

Divide each term in $2 y = 1$ by $2$ and simplify.

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Step 4.3.1

Divide each term in $2 y = 1$ by $2$ .

$\frac{2 y}{2} = \frac{1}{2}$

Step 4.3.2

Simplify the left side.

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Step 4.3.2.1

Cancel the common factor of $2$ .

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Step 4.3.2.1.1

Cancel the common factor.

$\frac{2 y}{2} = \frac{1}{2}$

Step 4.3.2.1.2

Divide $y$ by $1$ .

$y = \frac{1}{2}$