Basic Math Examples

$\frac{- 4 y + 3}{2} = \frac{7}{2 y}$

Step 1

Multiply the numerator of the first fraction by the denominator of the second fraction. Set this equal to the product of the denominator of the first fraction and the numerator of the second fraction.

$(- 4 y + 3) (2 y) = 2 \cdot 7$

Step 2

Solve the equation for $y$ .

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Step 2.1

Simplify.

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Step 2.1.1

Remove parentheses.

$(- 4 y + 3) \cdot 2 y = 2 \cdot 7$

Step 2.1.2

Multiply $2$ by $7$ .

$(- 4 y + 3) \cdot 2 y = 14$

Step 2.2

Divide each term in $(- 4 y + 3) \cdot (2 y) = 14$ by $2$ and simplify.

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Step 2.2.1

Divide each term in $(- 4 y + 3) \cdot (2 y) = 14$ by $2$ .

$\frac{(- 4 y + 3) \cdot (2 y)}{2} = \frac{14}{2}$

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Cancel the common factor of $2$ .

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Step 2.2.2.1.1

Cancel the common factor.

$\frac{(- 4 y + 3) \cdot (2 y)}{2} = \frac{14}{2}$

Step 2.2.2.1.2

Divide $(- 4 y + 3) \cdot (y)$ by $1$ .

$(- 4 y + 3) \cdot (y) = \frac{14}{2}$

Step 2.2.2.2

Apply the distributive property.

$- 4 y \cdot y + 3 y = \frac{14}{2}$

Step 2.2.2.3

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

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Step 2.2.2.3.1

Move $y$ .

$- 4 (y \cdot y) + 3 y = \frac{14}{2}$

Step 2.2.2.3.2

Multiply $y$ by $y$ .

$- 4 y^{2} + 3 y = \frac{14}{2}$

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Divide $14$ by $2$ .

$- 4 y^{2} + 3 y = 7$

Step 2.3

Subtract $7$ from both sides of the equation.

$- 4 y^{2} + 3 y - 7 = 0$

Step 2.4

Use the quadratic formula to find the solutions.

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

Step 2.5

Substitute the values $a = - 4$ , $b = 3$ , and $c = - 7$ into the quadratic formula and solve for $y$ .

$\frac{- 3 \pm \sqrt{3^{2} - 4 \cdot (- 4 \cdot - 7)}}{2 \cdot - 4}$

Step 2.6

Simplify.

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Step 2.6.1

Simplify the numerator.

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Step 2.6.1.1

Raise $3$ to the power of $2$ .

$y = \frac{- 3 \pm \sqrt{9 - 4 \cdot - 4 \cdot - 7}}{2 \cdot - 4}$

Step 2.6.1.2

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

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Step 2.6.1.2.1

Multiply $- 4$ by $- 4$ .

$y = \frac{- 3 \pm \sqrt{9 + 16 \cdot - 7}}{2 \cdot - 4}$

Step 2.6.1.2.2

Multiply $16$ by $- 7$ .

$y = \frac{- 3 \pm \sqrt{9 - 112}}{2 \cdot - 4}$

Step 2.6.1.3

Subtract $112$ from $9$ .

$y = \frac{- 3 \pm \sqrt{- 103}}{2 \cdot - 4}$

Step 2.6.1.4

Rewrite $- 103$ as $- 1 (103)$ .

$y = \frac{- 3 \pm \sqrt{- 1 \cdot 103}}{2 \cdot - 4}$

Step 2.6.1.5

Rewrite $\sqrt{- 1 (103)}$ as $\sqrt{- 1} \cdot \sqrt{103}$ .

$y = \frac{- 3 \pm \sqrt{- 1} \cdot \sqrt{103}}{2 \cdot - 4}$

Step 2.6.1.6

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

$y = \frac{- 3 \pm i \sqrt{103}}{2 \cdot - 4}$

Step 2.6.2

Multiply $2$ by $- 4$ .

$y = \frac{- 3 \pm i \sqrt{103}}{- 8}$

Step 2.6.3

Simplify $\frac{- 3 \pm i \sqrt{103}}{- 8}$ .

$y = \frac{3 \pm i \sqrt{103}}{8}$

Step 2.7

The final answer is the combination of both solutions.

$y = \frac{3 + i \sqrt{103}}{8}, \frac{3 - i \sqrt{103}}{8}$