Basic Math Examples

$7 x + 8 y \cdot (5 y) = 25$

Step 1

Simplify each term.

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Step 1.1

Rewrite using the commutative property of multiplication.

$7 x + 8 \cdot 5 y \cdot y = 25$

Step 1.2

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

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Step 1.2.1

Move $y$ .

$7 x + 8 \cdot 5 (y \cdot y) = 25$

Step 1.2.2

Multiply $y$ by $y$ .

$7 x + 8 \cdot 5 y^{2} = 25$

Step 1.3

Multiply $8$ by $5$ .

$7 x + 40 y^{2} = 25$

Step 2

Subtract $7 x$ from both sides of the equation.

$40 y^{2} = 25 - 7 x$

Step 3

Divide each term in $40 y^{2} = 25 - 7 x$ by $40$ and simplify.

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Step 3.1

Divide each term in $40 y^{2} = 25 - 7 x$ by $40$ .

$\frac{40 y^{2}}{40} = \frac{25}{40} + \frac{- 7 x}{40}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $40$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{40 y^{2}}{40} = \frac{25}{40} + \frac{- 7 x}{40}$

Step 3.2.1.2

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

$y^{2} = \frac{25}{40} + \frac{- 7 x}{40}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Simplify each term.

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Step 3.3.1.1

Cancel the common factor of $25$ and $40$ .

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Step 3.3.1.1.1

Factor $5$ out of $25$ .

$y^{2} = \frac{5 (5)}{40} + \frac{- 7 x}{40}$

Step 3.3.1.1.2

Cancel the common factors.

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Step 3.3.1.1.2.1

Factor $5$ out of $40$ .

$y^{2} = \frac{5 \cdot 5}{5 \cdot 8} + \frac{- 7 x}{40}$

Step 3.3.1.1.2.2

Cancel the common factor.

$y^{2} = \frac{5 \cdot 5}{5 \cdot 8} + \frac{- 7 x}{40}$

Step 3.3.1.1.2.3

Rewrite the expression.

$y^{2} = \frac{5}{8} + \frac{- 7 x}{40}$

Step 3.3.1.2

Move the negative in front of the fraction.

$y^{2} = \frac{5}{8} - \frac{7 x}{40}$

Step 4

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

$y = \pm \sqrt{\frac{5}{8} - \frac{7 x}{40}}$

Step 5

Simplify $\pm \sqrt{\frac{5}{8} - \frac{7 x}{40}}$ .

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Step 5.1

To write $\frac{5}{8}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$y = \pm \sqrt{\frac{5}{8} \cdot \frac{5}{5} - \frac{7 x}{40}}$

Step 5.2

Write each expression with a common denominator of $40$ , by multiplying each by an appropriate factor of $1$ .

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Step 5.2.1

Multiply $\frac{5}{8}$ by $\frac{5}{5}$ .

$y = \pm \sqrt{\frac{5 \cdot 5}{8 \cdot 5} - \frac{7 x}{40}}$

Step 5.2.2

Multiply $8$ by $5$ .

$y = \pm \sqrt{\frac{5 \cdot 5}{40} - \frac{7 x}{40}}$

Step 5.3

Combine the numerators over the common denominator.

$y = \pm \sqrt{\frac{5 \cdot 5 - 7 x}{40}}$

Step 5.4

Multiply $5$ by $5$ .

$y = \pm \sqrt{\frac{25 - 7 x}{40}}$

Step 5.5

Rewrite $\frac{25 - 7 x}{40}$ as ${(\frac{1}{2})}^{2} \frac{25 - 7 x}{10}$ .

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Step 5.5.1

Factor the perfect power $1^{2}$ out of $25 - 7 x$ .

$y = \pm \sqrt{\frac{1^{2} (25 - 7 x)}{40}}$

Step 5.5.2

Factor the perfect power $2^{2}$ out of $40$ .

$y = \pm \sqrt{\frac{1^{2} (25 - 7 x)}{2^{2} \cdot 10}}$

Step 5.5.3

Rearrange the fraction $\frac{1^{2} (25 - 7 x)}{2^{2} \cdot 10}$ .

$y = \pm \sqrt{{(\frac{1}{2})}^{2} \frac{25 - 7 x}{10}}$

Step 5.6

Pull terms out from under the radical.

$y = \pm \frac{1}{2} \sqrt{\frac{25 - 7 x}{10}}$

Step 5.7

Rewrite $\sqrt{\frac{25 - 7 x}{10}}$ as $\frac{\sqrt{25 - 7 x}}{\sqrt{10}}$ .

$y = \pm \frac{1}{2} \cdot \frac{\sqrt{25 - 7 x}}{\sqrt{10}}$

Step 5.8

Multiply $\frac{\sqrt{25 - 7 x}}{\sqrt{10}}$ by $\frac{\sqrt{10}}{\sqrt{10}}$ .

$y = \pm \frac{1}{2} (\frac{\sqrt{25 - 7 x}}{\sqrt{10}} \cdot \frac{\sqrt{10}}{\sqrt{10}})$

Step 5.9

Combine and simplify the denominator.

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Step 5.9.1

Multiply $\frac{\sqrt{25 - 7 x}}{\sqrt{10}}$ by $\frac{\sqrt{10}}{\sqrt{10}}$ .

$y = \pm \frac{1}{2} \cdot \frac{\sqrt{25 - 7 x} \sqrt{10}}{\sqrt{10} \sqrt{10}}$

Step 5.9.2

Raise $\sqrt{10}$ to the power of $1$ .

$y = \pm \frac{1}{2} \cdot \frac{\sqrt{25 - 7 x} \sqrt{10}}{{\sqrt{10}}^{1} \sqrt{10}}$

Step 5.9.3

Raise $\sqrt{10}$ to the power of $1$ .

$y = \pm \frac{1}{2} \cdot \frac{\sqrt{25 - 7 x} \sqrt{10}}{{\sqrt{10}}^{1} {\sqrt{10}}^{1}}$

Step 5.9.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y = \pm \frac{1}{2} \cdot \frac{\sqrt{25 - 7 x} \sqrt{10}}{{\sqrt{10}}^{1 + 1}}$

Step 5.9.5

Add $1$ and $1$ .

$y = \pm \frac{1}{2} \cdot \frac{\sqrt{25 - 7 x} \sqrt{10}}{{\sqrt{10}}^{2}}$

Step 5.9.6

Rewrite ${\sqrt{10}}^{2}$ as $10$ .

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Step 5.9.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{10}$ as $10^{\frac{1}{2}}$ .

$y = \pm \frac{1}{2} \cdot \frac{\sqrt{25 - 7 x} \sqrt{10}}{{(10^{\frac{1}{2}})}^{2}}$

Step 5.9.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$y = \pm \frac{1}{2} \cdot \frac{\sqrt{25 - 7 x} \sqrt{10}}{10^{\frac{1}{2} \cdot 2}}$

Step 5.9.6.3

Combine $\frac{1}{2}$ and $2$ .

$y = \pm \frac{1}{2} \cdot \frac{\sqrt{25 - 7 x} \sqrt{10}}{10^{\frac{2}{2}}}$

Step 5.9.6.4

Cancel the common factor of $2$ .

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Step 5.9.6.4.1

Cancel the common factor.

$y = \pm \frac{1}{2} \cdot \frac{\sqrt{25 - 7 x} \sqrt{10}}{10^{\frac{2}{2}}}$

Step 5.9.6.4.2

Rewrite the expression.

$y = \pm \frac{1}{2} \cdot \frac{\sqrt{25 - 7 x} \sqrt{10}}{10^{1}}$

Step 5.9.6.5

Evaluate the exponent.

$y = \pm \frac{1}{2} \cdot \frac{\sqrt{25 - 7 x} \sqrt{10}}{10}$

Step 5.10

Combine using the product rule for radicals.

$y = \pm \frac{1}{2} \cdot \frac{\sqrt{(25 - 7 x) \cdot 10}}{10}$

Step 5.11

Multiply $\frac{1}{2} \cdot \frac{\sqrt{(25 - 7 x) \cdot 10}}{10}$ .

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Step 5.11.1

Multiply $\frac{1}{2}$ by $\frac{\sqrt{(25 - 7 x) \cdot 10}}{10}$ .

$y = \pm \frac{\sqrt{(25 - 7 x) \cdot 10}}{2 \cdot 10}$

Step 5.11.2

Multiply $2$ by $10$ .

$y = \pm \frac{\sqrt{(25 - 7 x) \cdot 10}}{20}$

Step 5.12

Reorder factors in $\pm \frac{\sqrt{(25 - 7 x) \cdot 10}}{20}$ .

$y = \pm \frac{\sqrt{10 (25 - 7 x)}}{20}$

Step 6

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

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Step 6.1

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

$y = \frac{\sqrt{10 (25 - 7 x)}}{20}$

Step 6.2

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

$y = - \frac{\sqrt{10 (25 - 7 x)}}{20}$

Step 6.3

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

$y = \frac{\sqrt{10 (25 - 7 x)}}{20}$

$y = - \frac{\sqrt{10 (25 - 7 x)}}{20}$

$y = \frac{\sqrt{10 (25 - 7 x)}}{20}$

$y = - \frac{\sqrt{10 (25 - 7 x)}}{20}$