1. **State the problem:** We are given three points $(x, y)$ with $x$ values $-34$, $-17$, and $0$, and corresponding $y$ values $t$, $t + 29$, and $t + 58$, where $t$ is a constant. We need to find the linear equation $y = mx + b$ that fits these points. 2. **Recall the formula for a linear relationship:** The equation of a line is $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept. 3. **Calculate the slope $m$ using two points:** Using points $(-34, t)$ and $(-17, t + 29)$, $$m = \frac{(t + 29) - t}{-17 - (-34)} = \frac{29}{17}$$ 4. **Check the slope with the next pair of points:** Using points $(-17, t + 29)$ and $(0, t + 58)$, $$m = \frac{(t + 58) - (t + 29)}{0 - (-17)} = \frac{29}{17}$$ The slope is consistent. 5. **Find the y-intercept $b$:** Using point $(0, t + 58)$, $$y = mx + b \Rightarrow t + 58 = \frac{29}{17} \times 0 + b \Rightarrow b = t + 58$$ 6. **Write the equation:** $$y = \frac{29}{17} x + t + 58$$ 7. **Match with the options:** This corresponds to option (D). **Final answer:** $$\boxed{y = \frac{29}{17} x + t + 58}$$