Algebra Examples

$e^{x - 2} = e^{3 x + 1}$

Step 1

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

$e^{x - 2} = e^{3 x + 1}$

Step 2

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

$x - 2 = 3 x + 1$

Step 3

Solve for $x$ .

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Step 3.1

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

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Step 3.1.1

Subtract $3 x$ from both sides of the equation.

$x - 2 - 3 x = 1$

Step 3.1.2

Subtract $3 x$ from $x$ .

$- 2 x - 2 = 1$

Step 3.2

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

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Step 3.2.1

Add $2$ to both sides of the equation.

$- 2 x = 1 + 2$

Step 3.2.2

Add $1$ and $2$ .

$- 2 x = 3$

Step 3.3

Divide each term in $- 2 x = 3$ by $- 2$ and simplify.

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Step 3.3.1

Divide each term in $- 2 x = 3$ by $- 2$ .

$\frac{- 2 x}{- 2} = \frac{3}{- 2}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $- 2$ .

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Step 3.3.2.1.1

Cancel the common factor.

$\frac{- 2 x}{- 2} = \frac{3}{- 2}$

Step 3.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{3}{- 2}$

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

Move the negative in front of the fraction.

$x = - \frac{3}{2}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{3}{2}$

Decimal Form:

$x = - 1.5$

Mixed Number Form:

$x = - 1 \frac{1}{2}$