Trigonometry Examples

$25^{3 x - 1} = 5^{4 - 7 x}$

Step 1

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

$5^{2 (3 x - 1)} = 5^{4 - 7 x}$

Step 2

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

$2 (3 x - 1) = 4 - 7 x$

Step 3

Solve for $x$ .

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Step 3.1

Simplify $2 (3 x - 1)$ .

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Step 3.1.1

Rewrite.

$0 + 0 + 2 (3 x - 1) = 4 - 7 x$

Step 3.1.2

Simplify by adding zeros.

$2 (3 x - 1) = 4 - 7 x$

Step 3.1.3

Apply the distributive property.

$2 (3 x) + 2 \cdot - 1 = 4 - 7 x$

Step 3.1.4

Multiply.

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Step 3.1.4.1

Multiply $3$ by $2$ .

$6 x + 2 \cdot - 1 = 4 - 7 x$

Step 3.1.4.2

Multiply $2$ by $- 1$ .

$6 x - 2 = 4 - 7 x$

Step 3.2

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

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Step 3.2.1

Add $7 x$ to both sides of the equation.

$6 x - 2 + 7 x = 4$

Step 3.2.2

Add $6 x$ and $7 x$ .

$13 x - 2 = 4$

Step 3.3

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

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Step 3.3.1

Add $2$ to both sides of the equation.

$13 x = 4 + 2$

Step 3.3.2

Add $4$ and $2$ .

$13 x = 6$

Step 3.4

Divide each term in $13 x = 6$ by $13$ and simplify.

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Step 3.4.1

Divide each term in $13 x = 6$ by $13$ .

$\frac{13 x}{13} = \frac{6}{13}$

Step 3.4.2

Simplify the left side.

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Step 3.4.2.1

Cancel the common factor of $13$ .

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Step 3.4.2.1.1

Cancel the common factor.

$\frac{13 x}{13} = \frac{6}{13}$

Step 3.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{6}{13}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = \frac{6}{13}$

Decimal Form:

$x = 0.\overline{461538}$