Algebra Examples

$\frac{- 12}{y} = y - 7$

Step 1

Move the negative in front of the fraction.

$- \frac{12}{y} = y - 7$

Step 2

Find the LCD of the terms in the equation.

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Step 2.1

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

$y, 1, 1$

Step 2.2

The LCM of one and any expression is the expression.

$y$

Step 3

Multiply each term in $- \frac{12}{y} = y - 7$ by $y$ to eliminate the fractions.

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Step 3.1

Multiply each term in $- \frac{12}{y} = y - 7$ by $y$ .

$- \frac{12}{y} y = y \cdot y - 7 y$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $y$ .

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Step 3.2.1.1

Move the leading negative in $- \frac{12}{y}$ into the numerator.

$\frac{- 12}{y} y = y \cdot y - 7 y$

Step 3.2.1.2

Cancel the common factor.

$\frac{- 12}{y} y = y \cdot y - 7 y$

Step 3.2.1.3

Rewrite the expression.

$- 12 = y \cdot y - 7 y$

Step 3.3

Simplify the right side.

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Step 3.3.1

Multiply $y$ by $y$ .

$- 12 = y^{2} - 7 y$

Step 4

Solve the equation.

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Step 4.1

Rewrite the equation as $y^{2} - 7 y = - 12$ .

$y^{2} - 7 y = - 12$

Step 4.2

Add $12$ to both sides of the equation.

$y^{2} - 7 y + 12 = 0$

Step 4.3

Factor $y^{2} - 7 y + 12$ using the AC method.

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Step 4.3.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $12$ and whose sum is $- 7$ .

$- 4, - 3$

Step 4.3.2

Write the factored form using these integers.

$(y - 4) (y - 3) = 0$

Step 4.4

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$y - 4 = 0$

$y - 3 = 0$

Step 4.5

Set $y - 4$ equal to $0$ and solve for $y$ .

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Step 4.5.1

Set $y - 4$ equal to $0$ .

$y - 4 = 0$

Step 4.5.2

Add $4$ to both sides of the equation.

$y = 4$

Step 4.6

Set $y - 3$ equal to $0$ and solve for $y$ .

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Step 4.6.1

Set $y - 3$ equal to $0$ .

$y - 3 = 0$

Step 4.6.2

Add $3$ to both sides of the equation.

$y = 3$

Step 4.7

The final solution is all the values that make $(y - 4) (y - 3) = 0$ true.

$y = 4, 3$