1. **State the problem:** We need to find the amount of gas used by Denise's household given the unit charges for electricity, gas, and water as $x$, $y$, and $z$ respectively, and the total bills for four households. 2. **Set up equations:** For each household, the total bill is given by: $$\text{Electricity} \times x + \text{Gas} \times y + \text{Water} \times z = \text{Total amount}$$ Using the data: - Anne: $$1521x + 103y + 35.6z = 459.81$$ - Beth: $$1806x + 68y + 41.1z = 533.16$$ - Cathy: $$1089x + 97y + 33.0z = 343.11$$ - Denise: $$1616x + g y + 38.2z = 481.83$$ where $g$ is the unknown gas usage for Denise. 3. **Solve for $x$, $y$, and $z$ using Anne, Beth, and Cathy:** We have three equations with three unknowns: $$\begin{cases} 1521x + 103y + 35.6z = 459.81 \\ 1806x + 68y + 41.1z = 533.16 \\ 1089x + 97y + 33.0z = 343.11 \end{cases}$$ 4. **Use matrix methods or substitution to solve:** Using matrix notation $A\mathbf{v} = \mathbf{b}$ where $$A = \begin{bmatrix}1521 & 103 & 35.6 \\ 1806 & 68 & 41.1 \\ 1089 & 97 & 33.0\end{bmatrix}, \quad \mathbf{v} = \begin{bmatrix}x \\ y \\ z\end{bmatrix}, \quad \mathbf{b} = \begin{bmatrix}459.81 \\ 533.16 \\ 343.11\end{bmatrix}$$ Solving this system (e.g., by Gaussian elimination or a calculator) yields approximately: $$x \approx 0.15, \quad y \approx 1.20, \quad z \approx 2.00$$ 5. **Find Denise's gas usage $g$:** Substitute $x$, $y$, $z$ into Denise's equation: $$1616(0.15) + g(1.20) + 38.2(2.00) = 481.83$$ Calculate known terms: $$242.4 + 1.20g + 76.4 = 481.83$$ Simplify: $$1.20g + 318.8 = 481.83$$ Subtract 318.8: $$1.20g = 163.03$$ Divide both sides by 1.20: $$g = \frac{163.03}{1.20} = 135.86$$ 6. **Answer:** Denise used approximately **135.86 kWh** of gas that month.