1. **State the problem:** We are given two equations: $$1.2x = 700 \cdot 1.034^y$$ $$x = 116.5 \cdot 1.023^y$$ We need to find the value of $y$. 2. **Substitute $x$ from the second equation into the first:** $$1.2 \times \left(116.5 \cdot 1.023^y\right) = 700 \cdot 1.034^y$$ 3. **Simplify the left side:** $$1.2 \times 116.5 \cdot 1.023^y = 700 \cdot 1.034^y$$ $$139.8 \cdot 1.023^y = 700 \cdot 1.034^y$$ 4. **Divide both sides by $1.023^y$ to isolate terms with $y$ on one side:** $$\frac{139.8 \cdot \cancel{1.023^y}}{\cancel{1.023^y}} = \frac{700 \cdot 1.034^y}{1.023^y}$$ $$139.8 = 700 \cdot \left(\frac{1.034}{1.023}\right)^y$$ 5. **Divide both sides by 700:** $$\frac{139.8}{700} = \left(\frac{1.034}{1.023}\right)^y$$ $$0.1997 = \left(1.01074\right)^y$$ 6. **Take the natural logarithm of both sides:** $$\ln(0.1997) = \ln\left(1.01074^y\right)$$ $$\ln(0.1997) = y \cdot \ln(1.01074)$$ 7. **Solve for $y$:** $$y = \frac{\ln(0.1997)}{\ln(1.01074)}$$ Calculate the values: $$\ln(0.1997) \approx -1.610$$ $$\ln(1.01074) \approx 0.01069$$ $$y \approx \frac{-1.610}{0.01069} \approx -150.6$$ **Final answer:** $$y \approx -150.6$$