6

4(a) (i) Solve $2 \mathrm{e}^{1-2 y}=3 \mathrm{e}^{3 y+2}$. Give your answer in exact form.

Q: 6 4(a) (i) Solve $2 \mathrm{e}^{1-2 y}=3 \mathrm{e}^{3 y+2}$. Give your answer in exact form.

To solve the equation $2 e^{1 - 2y} = 3 e^{3y + 2}$, we can start by simplifying it step by step. 1. **Rearranging the equation**: Divide both sides by $e^{1 - 2y}$: \[ 2 = 3 e^{3y + 2} / e^{1 - 2y} \] This simplifies to: \[ 2 = 3 e^{(3y + 2) - (1 - 2y)} = 3 e^{5y + 1} \] 2. **Isolating the exponential**: Now, we can isolate $e^{5y + 1}$: \[ e^{5y + 1} = \frac{2}{3} \] 3. **Taking the natural logarithm**: To solve for $y$, take the natural logarithm of both sides: \[ 5y + 1 = \ln\left(\frac{2}{3}\right) \] 4. **Solving for $y$**: Now, isolate $y$: \[ 5y = \ln\left(\frac{2}{3}\right) - 1 \] \[ y = \frac{\ln\left(\frac{2}{3}\right) - 1}{5} \] So, the exact form of the solution is: \[ y = \frac{\ln\left(\frac{2}{3}\right) - 1}{5} \] If you have any further questions or need clarification on any steps, feel free to ask!

Question

6

4(a) (i) Solve $2 \mathrm{e}^{1-2 y}=3 \mathrm{e}^{3 y+2}$. Give your answer in exact form.

Accepted Answer

To solve the equation $2 e^{1 - 2y} = 3 e^{3y + 2}$, we can start by simplifying it step by step. 1. **Rearranging the equation**: Divide both sides by $e^{1 - 2y}$: $$ 2 = 3 e^{3y + 2} / e^{1 - 2y} $$ This simplifies to: $$ 2 = 3 e^{(3y + 2) - (1 - 2y)} = 3 e^{5y + 1} $$ 2. **Isolating the exponential**: Now, we can isolate $e^{5y + 1}$: $$ e^{5y + 1} = \frac{2}{3} $$ 3. **Taking the natural logarithm**: To solve for $y$, take the natural logarithm of both sides: $$ 5y + 1 = \ln\left(\frac{2}{3}\right) $$ 4. **Solving for $y$**: Now, isolate $y$: $$ 5y = \ln\left(\frac{2}{3}\right) - 1 $$ $$ y = \frac{\ln\left(\frac{2}{3}\right) - 1}{5} $$ So, the exact form of the solution is: $$ y = \frac{\ln\left(\frac{2}{3}\right) - 1}{5} $$ If you have any further questions or need clarification on any steps, feel free to ask!

Sia