Algebra Examples

$5 = \frac{3 + \sqrt{9 - 4 y}}{2}$

Step 1

Move all terms to the left side of the equation and simplify.

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Step 1.1

Simplify the right side.

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Step 1.1.1

Split the fraction $\frac{3 + \sqrt{9 - 4 y}}{2}$ into two fractions.

$5 = \frac{3}{2} + \frac{\sqrt{9 - 4 y}}{2}$

Step 1.2

Move all the expressions to the left side of the equation.

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Step 1.2.1

Subtract $\frac{3}{2}$ from both sides of the equation.

$5 - \frac{3}{2} = \frac{\sqrt{9 - 4 y}}{2}$

Step 1.2.2

Subtract $\frac{\sqrt{9 - 4 y}}{2}$ from both sides of the equation.

$5 - \frac{3}{2} - \frac{\sqrt{9 - 4 y}}{2} = 0$

Step 1.3

Simplify $5 - \frac{3}{2} - \frac{\sqrt{9 - 4 y}}{2}$ .

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Step 1.3.1

To write $5$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$5 \cdot \frac{2}{2} - \frac{3}{2} - \frac{\sqrt{9 - 4 y}}{2} = 0$

Step 1.3.2

Combine $5$ and $\frac{2}{2}$ .

$\frac{5 \cdot 2}{2} - \frac{3}{2} - \frac{\sqrt{9 - 4 y}}{2} = 0$

Step 1.3.3

Combine the numerators over the common denominator.

$\frac{5 \cdot 2 - 3}{2} - \frac{\sqrt{9 - 4 y}}{2} = 0$

Step 1.3.4

Simplify the numerator.

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Step 1.3.4.1

Multiply $5$ by $2$ .

$\frac{10 - 3}{2} - \frac{\sqrt{9 - 4 y}}{2} = 0$

Step 1.3.4.2

Subtract $3$ from $10$ .

$\frac{7}{2} - \frac{\sqrt{9 - 4 y}}{2} = 0$

Step 2

Solve for $\sqrt{9 - 4 y}$ .

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Step 2.1

Subtract $\frac{7}{2}$ from both sides of the equation.

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

Step 2.2

Since the expression on each side of the equation has the same denominator, the numerators must be equal.

$- \sqrt{9 - 4 y} = - 7$

Step 2.3

Divide each term in $- \sqrt{9 - 4 y} = - 7$ by $- 1$ and simplify.

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Step 2.3.1

Divide each term in $- \sqrt{9 - 4 y} = - 7$ by $- 1$ .

$\frac{- \sqrt{9 - 4 y}}{- 1} = \frac{- 7}{- 1}$

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Dividing two negative values results in a positive value.

$\frac{\sqrt{9 - 4 y}}{1} = \frac{- 7}{- 1}$

Step 2.3.2.2

Divide $\sqrt{9 - 4 y}$ by $1$ .

$\sqrt{9 - 4 y} = \frac{- 7}{- 1}$

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

Divide $- 7$ by $- 1$ .

$\sqrt{9 - 4 y} = 7$

Step 3

To remove the radical on the left side of the equation, square both sides of the equation.

${\sqrt{9 - 4 y}}^{2} = 7^{2}$

Step 4

Simplify each side of the equation.

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Step 4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{9 - 4 y}$ as ${(9 - 4 y)}^{\frac{1}{2}}$ .

${({(9 - 4 y)}^{\frac{1}{2}})}^{2} = 7^{2}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Simplify ${({(9 - 4 y)}^{\frac{1}{2}})}^{2}$ .

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Step 4.2.1.1

Multiply the exponents in ${({(9 - 4 y)}^{\frac{1}{2}})}^{2}$ .

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Step 4.2.1.1.1

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

${(9 - 4 y)}^{\frac{1}{2} \cdot 2} = 7^{2}$

Step 4.2.1.1.2

Cancel the common factor of $2$ .

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Step 4.2.1.1.2.1

Cancel the common factor.

${(9 - 4 y)}^{\frac{1}{2} \cdot 2} = 7^{2}$

Step 4.2.1.1.2.2

Rewrite the expression.

${(9 - 4 y)}^{1} = 7^{2}$

Step 4.2.1.2

Simplify.

$9 - 4 y = 7^{2}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Raise $7$ to the power of $2$ .

$9 - 4 y = 49$

Step 5

Solve for $y$ .

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Step 5.1

Move all terms not containing $y$ to the right side of the equation.

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Step 5.1.1

Subtract $9$ from both sides of the equation.

$- 4 y = 49 - 9$

Step 5.1.2

Subtract $9$ from $49$ .

$- 4 y = 40$

Step 5.2

Divide each term in $- 4 y = 40$ by $- 4$ and simplify.

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Step 5.2.1

Divide each term in $- 4 y = 40$ by $- 4$ .

$\frac{- 4 y}{- 4} = \frac{40}{- 4}$

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Cancel the common factor of $- 4$ .

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Step 5.2.2.1.1

Cancel the common factor.

$\frac{- 4 y}{- 4} = \frac{40}{- 4}$

Step 5.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{40}{- 4}$

Step 5.2.3

Simplify the right side.

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Step 5.2.3.1

Divide $40$ by $- 4$ .

$y = - 10$