Algebra Examples

$3^{2 y - 2} = 81^{2 y + 1}$

Step 1

Create equivalent expressions in the equation that all have equal bases.

$3^{2 y - 2} = 3^{4 (2 y + 1)}$

Step 2

Since the bases are the same, then two expressions are only equal if the exponents are also equal.

$2 y - 2 = 4 (2 y + 1)$

Step 3

Solve for $y$ .

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Step 3.1

Simplify $4 (2 y + 1)$ .

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Step 3.1.1

Rewrite.

$2 y - 2 = 0 + 0 + 4 (2 y + 1)$

Step 3.1.2

Simplify by adding zeros.

$2 y - 2 = 4 (2 y + 1)$

Step 3.1.3

Apply the distributive property.

$2 y - 2 = 4 (2 y) + 4 \cdot 1$

Step 3.1.4

Multiply.

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Step 3.1.4.1

Multiply $2$ by $4$ .

$2 y - 2 = 8 y + 4 \cdot 1$

Step 3.1.4.2

Multiply $4$ by $1$ .

$2 y - 2 = 8 y + 4$

Step 3.2

Move all terms containing $y$ to the left side of the equation.

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Step 3.2.1

Subtract $8 y$ from both sides of the equation.

$2 y - 2 - 8 y = 4$

Step 3.2.2

Subtract $8 y$ from $2 y$ .

$- 6 y - 2 = 4$

Step 3.3

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

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Step 3.3.1

Add $2$ to both sides of the equation.

$- 6 y = 4 + 2$

Step 3.3.2

Add $4$ and $2$ .

$- 6 y = 6$

Step 3.4

Divide each term in $- 6 y = 6$ by $- 6$ and simplify.

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Step 3.4.1

Divide each term in $- 6 y = 6$ by $- 6$ .

$\frac{- 6 y}{- 6} = \frac{6}{- 6}$

Step 3.4.2

Simplify the left side.

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Step 3.4.2.1

Cancel the common factor of $- 6$ .

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Step 3.4.2.1.1

Cancel the common factor.

$\frac{- 6 y}{- 6} = \frac{6}{- 6}$

Step 3.4.2.1.2

Divide $y$ by $1$ .

$y = \frac{6}{- 6}$

Step 3.4.3

Simplify the right side.

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Step 3.4.3.1

Divide $6$ by $- 6$ .

$y = - 1$