Algebra Examples

$f (x) = - \frac{1}{3} \cdot 6^{x} + 5$

Step 1

Find the x-intercepts.

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Step 1.1

To find the x-intercept(s), substitute in $0$ for $y$ and solve for $x$ .

$0 = - \frac{1}{3} \cdot 6^{x} + 5$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $- \frac{1}{3} \cdot 6^{x} + 5 = 0$ .

$- \frac{1}{3} \cdot 6^{x} + 5 = 0$

Step 1.2.2

Combine $6^{x}$ and $\frac{1}{3}$ .

$- \frac{6^{x}}{3} + 5 = 0$

Step 1.2.3

Subtract $5$ from both sides of the equation.

$- \frac{6^{x}}{3} = - 5$

Step 1.2.4

Multiply both sides of the equation by $- 3$ .

$- 3 (- \frac{6^{x}}{3}) = - 3 \cdot - 5$

Step 1.2.5

Simplify both sides of the equation.

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Step 1.2.5.1

Simplify the left side.

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Step 1.2.5.1.1

Simplify $- 3 (- \frac{6^{x}}{3})$ .

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Step 1.2.5.1.1.1

Cancel the common factor of $3$ .

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Step 1.2.5.1.1.1.1

Move the leading negative in $- \frac{6^{x}}{3}$ into the numerator.

$- 3 \frac{- 6^{x}}{3} = - 3 \cdot - 5$

Step 1.2.5.1.1.1.2

Factor $3$ out of $- 3$ .

$3 (- 1) \frac{- 6^{x}}{3} = - 3 \cdot - 5$

Step 1.2.5.1.1.1.3

Cancel the common factor.

$3 \cdot - 1 \frac{- 6^{x}}{3} = - 3 \cdot - 5$

Step 1.2.5.1.1.1.4

Rewrite the expression.

$- - 6^{x} = - 3 \cdot - 5$

Step 1.2.5.1.1.2

Multiply.

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Step 1.2.5.1.1.2.1

Multiply $- 1$ by $- 1$ .

$1 \cdot 6^{x} = - 3 \cdot - 5$

Step 1.2.5.1.1.2.2

Multiply $6^{x}$ by $1$ .

$6^{x} = - 3 \cdot - 5$

Step 1.2.5.2

Simplify the right side.

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Step 1.2.5.2.1

Multiply $- 3$ by $- 5$ .

$6^{x} = 15$

Step 1.2.6

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

$\ln (6^{x}) = \ln (15)$

Step 1.2.7

Expand $\ln (6^{x})$ by moving $x$ outside the logarithm.

$x \ln (6) = \ln (15)$

Step 1.2.8

Divide each term in $x \ln (6) = \ln (15)$ by $\ln (6)$ and simplify.

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Step 1.2.8.1

Divide each term in $x \ln (6) = \ln (15)$ by $\ln (6)$ .

$\frac{x \ln (6)}{\ln (6)} = \frac{\ln (15)}{\ln (6)}$

Step 1.2.8.2

Simplify the left side.

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Step 1.2.8.2.1

Cancel the common factor of $\ln (6)$ .

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Step 1.2.8.2.1.1

Cancel the common factor.

$\frac{x \ln (6)}{\ln (6)} = \frac{\ln (15)}{\ln (6)}$

Step 1.2.8.2.1.2

Divide $x$ by $1$ .

$x = \frac{\ln (15)}{\ln (6)}$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(\frac{\ln (15)}{\ln (6)}, 0)$

Step 2

Find the y-intercepts.

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Step 2.1

To find the y-intercept(s), substitute in $0$ for $x$ and solve for $y$ .

$y = - \frac{1}{3} \cdot 6^{0} + 5$

Step 2.2

Simplify $- \frac{1}{3} \cdot 6^{0} + 5$ .

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Step 2.2.1

Simplify each term.

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Step 2.2.1.1

Anything raised to $0$ is $1$ .

$y = - \frac{1}{3} \cdot 1 + 5$

Step 2.2.1.2

Multiply $- 1$ by $1$ .

$y = - \frac{1}{3} + 5$

Step 2.2.2

To write $5$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$y = - \frac{1}{3} + 5 \cdot \frac{3}{3}$

Step 2.2.3

Combine $5$ and $\frac{3}{3}$ .

$y = - \frac{1}{3} + \frac{5 \cdot 3}{3}$

Step 2.2.4

Combine the numerators over the common denominator.

$y = \frac{- 1 + 5 \cdot 3}{3}$

Step 2.2.5

Simplify the numerator.

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Step 2.2.5.1

Multiply $5$ by $3$ .

$y = \frac{- 1 + 15}{3}$

Step 2.2.5.2

Add $- 1$ and $15$ .

$y = \frac{14}{3}$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, \frac{14}{3})$

Step 3

List the intersections.

x-intercept(s): $(\frac{\ln (15)}{\ln (6)}, 0)$

y-intercept(s): $(0, \frac{14}{3})$

Step 4