1. **State the problem:** We are given two points $(15,10)$ and $(30,15)$ and need to find the value of $y$ when $x=17.9$ using linear interpolation. 2. **Formula for linear interpolation:** $$y = y_1 + \frac{(x - x_1)(y_2 - y_1)}{x_2 - x_1}$$ where $(x_1,y_1) = (15,10)$ and $(x_2,y_2) = (30,15)$. 3. **Substitute the known values:** $$y = 10 + \frac{(17.9 - 15)(15 - 10)}{30 - 15}$$ 4. **Calculate the differences:** $$y = 10 + \frac{2.9 \times 5}{15}$$ 5. **Simplify the fraction:** $$y = 10 + \frac{2.9 \times 5}{\cancel{15}} = 10 + \frac{2.9 \times 5}{\cancel{15}}$$ Since $15 = 3 \times 5$, we can cancel $5$: $$y = 10 + \frac{2.9 \times \cancel{5}}{3 \times \cancel{5}} = 10 + \frac{2.9}{3}$$ 6. **Evaluate the fraction:** $$\frac{2.9}{3} \approx 0.9667$$ 7. **Add to 10:** $$y \approx 10 + 0.9667 = 10.9667$$ **Final answer:** $$y \approx 10.97$$ This means when $x=17.9$, the interpolated value of $y$ is approximately $10.97$.