1. **Stating the problem:** We are given a table of values for $x$, $y_1$, and $y_2$. The goal is to analyze or interpret the data for $y_1$ and $y_2$ as functions of $x$. 2. **Observing the data:** - $x$ increases by 2 each step: 2, 4, 6, 8, 10, 12. - $y_1$ values increase steadily: 41, 87, 133, 179, 225, 271. - $y_2$ values decrease steadily: 110, 96, 82, 68, 54, 40. 3. **Finding the formulas for $y_1$ and $y_2$ as linear functions of $x$:** Assuming linear relationships: $$y_1 = m_1 x + b_1$$ $$y_2 = m_2 x + b_2$$ 4. **Calculate slope $m_1$ for $y_1$:** Using points $(2,41)$ and $(4,87)$: $$m_1 = \frac{87 - 41}{4 - 2} = \frac{46}{2} = 23$$ 5. **Calculate intercept $b_1$ for $y_1$:** Using point $(2,41)$: $$41 = 23 \times 2 + b_1 \Rightarrow b_1 = 41 - 46 = -5$$ 6. **Formula for $y_1$:** $$y_1 = 23x - 5$$ 7. **Calculate slope $m_2$ for $y_2$:** Using points $(2,110)$ and $(4,96)$: $$m_2 = \frac{96 - 110}{4 - 2} = \frac{-14}{2} = -7$$ 8. **Calculate intercept $b_2$ for $y_2$:** Using point $(2,110)$: $$110 = -7 \times 2 + b_2 \Rightarrow b_2 = 110 + 14 = 124$$ 9. **Formula for $y_2$:** $$y_2 = -7x + 124$$ 10. **Verification:** Check $y_1$ at $x=6$: $$23 \times 6 - 5 = 138 - 5 = 133$$ matches table. Check $y_2$ at $x=6$: $$-7 \times 6 + 124 = -42 + 124 = 82$$ matches table. **Final answer:** $$y_1 = 23x - 5$$ $$y_2 = -7x + 124$$