Find the x and y Intercepts 3y=-5^x+7

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

Step 1

Find the x-intercepts.

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

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

Solve the equation.

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

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

Multiply $3$ by $0$ .

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

Subtract $7$ from both sides of the equation.

$- 5^{x} = - 7$

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

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

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

Simplify the left side.

Dividing two negative values results in a positive value.

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

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

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

Simplify the right side.

Divide $- 7$ by $- 1$ .

$5^{x} = 7$

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

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

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

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

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

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

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

Simplify the left side.

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

Cancel the common factor.

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

Divide $x$ by $1$ .

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

x-intercept(s) in point form.

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

Step 2

Find the y-intercepts.

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

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

Solve the equation.

Simplify.

Anything raised to $0$ is $1$ .

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

Multiply $- 1$ by $1$ .

$3 y = - 1 + 7$

Add $- 1$ and $7$ .

$3 y = 6$

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

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

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

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

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

Divide $y$ by $1$ .

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

Simplify the right side.

Divide $6$ by $3$ .

$y = 2$

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

Find the x and y Intercepts 3y=-5^x+7

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