Algebra Examples

$3 y = - 5^{x} + 7$

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 $x$ .

$3 (0) = - 5^{x} + 7$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $- 5^{x} + 7 = 3 (0)$ .

$- 5^{x} + 7 = 3 (0)$

Step 1.2.2

Multiply $3$ by $0$ .

$- 5^{x} + 7 = 0$

Step 1.2.3

Subtract $7$ from both sides of the equation.

$- 5^{x} = - 7$

Step 1.2.4

Divide each term in $- 5^{x} = - 7$ by $- 1$ and simplify.

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Step 1.2.4.1

Divide each term in $- 5^{x} = - 7$ by $- 1$ .

$\frac{- 5^{x}}{- 1} = \frac{- 7}{- 1}$

Step 1.2.4.2

Simplify the left side.

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Step 1.2.4.2.1

Dividing two negative values results in a positive value.

$\frac{5^{x}}{1} = \frac{- 7}{- 1}$

Step 1.2.4.2.2

Divide $5^{x}$ by $1$ .

$5^{x} = \frac{- 7}{- 1}$

Step 1.2.4.3

Simplify the right side.

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Step 1.2.4.3.1

Divide $- 7$ by $- 1$ .

$5^{x} = 7$

Step 1.2.5

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

$\ln (5^{x}) = \ln (7)$

Step 1.2.6

Expand $\ln (5^{x})$ by moving $x$ outside the logarithm.

$x \ln (5) = \ln (7)$

Step 1.2.7

Divide each term in $x \ln (5) = \ln (7)$ by $\ln (5)$ and simplify.

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Step 1.2.7.1

Divide each term in $x \ln (5) = \ln (7)$ by $\ln (5)$ .

$\frac{x \ln (5)}{\ln (5)} = \frac{\ln (7)}{\ln (5)}$

Step 1.2.7.2

Simplify the left side.

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Step 1.2.7.2.1

Cancel the common factor of $\ln (5)$ .

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Step 1.2.7.2.1.1

Cancel the common factor.

$\frac{x \ln (5)}{\ln (5)} = \frac{\ln (7)}{\ln (5)}$

Step 1.2.7.2.1.2

Divide $x$ by $1$ .

$x = \frac{\ln (7)}{\ln (5)}$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(\frac{\ln (7)}{\ln (5)}, 0)$

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 $y$ .

$3 y = - 5^{0} + 7$

Step 2.2

Solve the equation.

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Step 2.2.1

Simplify.

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Step 2.2.1.1

Anything raised to $0$ is $1$ .

$3 y = - 1 \cdot 1 + 7$

Step 2.2.1.2

Multiply $- 1$ by $1$ .

$3 y = - 1 + 7$

Step 2.2.1.3

Add $- 1$ and $7$ .

$3 y = 6$

Step 2.2.2

Divide each term in $3 y = 6$ by $3$ and simplify.

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Step 2.2.2.1

Divide each term in $3 y = 6$ by $3$ .

$\frac{3 y}{3} = \frac{6}{3}$

Step 2.2.2.2

Simplify the left side.

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Step 2.2.2.2.1

Cancel the common factor of $3$ .

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Step 2.2.2.2.1.1

Cancel the common factor.

$\frac{3 y}{3} = \frac{6}{3}$

Step 2.2.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{6}{3}$

Step 2.2.2.3

Simplify the right side.

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Step 2.2.2.3.1

Divide $6$ by $3$ .

$y = 2$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, 2)$

Step 3

List the intersections.

x-intercept(s): $(\frac{\ln (7)}{\ln (5)}, 0)$

y-intercept(s): $(0, 2)$

Step 4