Basic Math Examples

$21 \div 6 y = 9 y - 12$

Step 1

Cancel the common factor of $21$ and $6$ .

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Step 1.1

Factor $3$ out of $21$ .

$3 (7) \div 6 y = 9 y - 12$

Step 1.2

Cancel the common factors.

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Step 1.2.1

Factor $3$ out of $6 y$ .

$3 (7) \div 3 (2 y) = 9 y - 12$

Step 1.2.2

Cancel the common factor.

$3 \cdot 7 \div 3 (2 y) = 9 y - 12$

Step 1.2.3

Rewrite the expression.

$\frac{7}{2 y} = 9 y - 12$

Step 2

Multiply both sides by $2 y$ .

$\frac{7}{2 y} (2 y) = (9 y - 12) (2 y)$

Step 3

Simplify.

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Step 3.1

Simplify the left side.

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Step 3.1.1

Simplify $\frac{7}{2 y} (2 y)$ .

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Step 3.1.1.1

Rewrite using the commutative property of multiplication.

$2 \frac{7}{2 y} y = (9 y - 12) (2 y)$

Step 3.1.1.2

Cancel the common factor of $2$ .

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Step 3.1.1.2.1

Factor $2$ out of $2 y$ .

$2 \frac{7}{2 (y)} y = (9 y - 12) (2 y)$

Step 3.1.1.2.2

Cancel the common factor.

$2 \frac{7}{2 y} y = (9 y - 12) (2 y)$

Step 3.1.1.2.3

Rewrite the expression.

$\frac{7}{y} y = (9 y - 12) (2 y)$

Step 3.1.1.3

Cancel the common factor of $y$ .

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Step 3.1.1.3.1

Cancel the common factor.

$\frac{7}{y} y = (9 y - 12) (2 y)$

Step 3.1.1.3.2

Rewrite the expression.

$7 = (9 y - 12) (2 y)$

Step 3.2

Simplify the right side.

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Step 3.2.1

Simplify $(9 y - 12) (2 y)$ .

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Step 3.2.1.1

Simplify by multiplying through.

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Step 3.2.1.1.1

Apply the distributive property.

$7 = 9 y (2 y) - 12 (2 y)$

Step 3.2.1.1.2

Simplify the expression.

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Step 3.2.1.1.2.1

Rewrite using the commutative property of multiplication.

$7 = 9 \cdot 2 y \cdot y - 12 (2 y)$

Step 3.2.1.1.2.2

Multiply $2$ by $- 12$ .

$7 = 9 \cdot 2 y \cdot y - 24 y$

Step 3.2.1.2

Simplify each term.

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Step 3.2.1.2.1

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

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Step 3.2.1.2.1.1

Move $y$ .

$7 = 9 \cdot 2 (y \cdot y) - 24 y$

Step 3.2.1.2.1.2

Multiply $y$ by $y$ .

$7 = 9 \cdot 2 y^{2} - 24 y$

Step 3.2.1.2.2

Multiply $9$ by $2$ .

$7 = 18 y^{2} - 24 y$

Step 4

Solve for $y$ .

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Step 4.1

Rewrite the equation as $18 y^{2} - 24 y = 7$ .

$18 y^{2} - 24 y = 7$

Step 4.2

Subtract $7$ from both sides of the equation.

$18 y^{2} - 24 y - 7 = 0$

Step 4.3

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 4.4

Substitute the values $a = 18$ , $b = - 24$ , and $c = - 7$ into the quadratic formula and solve for $y$ .

$\frac{24 \pm \sqrt{{(- 24)}^{2} - 4 \cdot (18 \cdot - 7)}}{2 \cdot 18}$

Step 4.5

Simplify.

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Step 4.5.1

Simplify the numerator.

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Step 4.5.1.1

Raise $- 24$ to the power of $2$ .

$y = \frac{24 \pm \sqrt{576 - 4 \cdot 18 \cdot - 7}}{2 \cdot 18}$

Step 4.5.1.2

Multiply $- 4 \cdot 18 \cdot - 7$ .

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Step 4.5.1.2.1

Multiply $- 4$ by $18$ .

$y = \frac{24 \pm \sqrt{576 - 72 \cdot - 7}}{2 \cdot 18}$

Step 4.5.1.2.2

Multiply $- 72$ by $- 7$ .

$y = \frac{24 \pm \sqrt{576 + 504}}{2 \cdot 18}$

Step 4.5.1.3

Add $576$ and $504$ .

$y = \frac{24 \pm \sqrt{1080}}{2 \cdot 18}$

Step 4.5.1.4

Rewrite $1080$ as $6^{2} \cdot 30$ .

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Step 4.5.1.4.1

Factor $36$ out of $1080$ .

$y = \frac{24 \pm \sqrt{36 (30)}}{2 \cdot 18}$

Step 4.5.1.4.2

Rewrite $36$ as $6^{2}$ .

$y = \frac{24 \pm \sqrt{6^{2} \cdot 30}}{2 \cdot 18}$

Step 4.5.1.5

Pull terms out from under the radical.

$y = \frac{24 \pm 6 \sqrt{30}}{2 \cdot 18}$

Step 4.5.2

Multiply $2$ by $18$ .

$y = \frac{24 \pm 6 \sqrt{30}}{36}$

Step 4.5.3

Simplify $\frac{24 \pm 6 \sqrt{30}}{36}$ .

$y = \frac{4 \pm \sqrt{30}}{6}$

Step 4.6

The final answer is the combination of both solutions.

$y = \frac{4 + \sqrt{30}}{6}, \frac{4 - \sqrt{30}}{6}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$y = \frac{4 + \sqrt{30}}{6}, \frac{4 - \sqrt{30}}{6}$

Decimal Form:

$y = 1.57953759 \dots, - 0.24620426 \dots$