1. **State the problem:** We are given the amount of money Francis has in his bank account over several days and need to find a rule for the amount of money $y$ in terms of days $x$. 2. **Analyze the data:** The table shows: - Day 0: $27$ - Day 1: $70$ - Day 2: $113$ - Day 3: $156$ 3. **Find the pattern:** Calculate the daily increase: $$70 - 27 = 43$$ $$113 - 70 = 43$$ $$156 - 113 = 43$$ The amount increases by $43$ each day. 4. **Write the rule:** Since the amount increases by a constant $43$ each day, the relationship is linear: $$y = mx + b$$ where $m$ is the rate of change (slope) and $b$ is the initial amount. 5. **Identify $m$ and $b$:** - $m = 43$ - $b = 27$ (amount at day 0) So the rule is: $$y = 43x + 27$$ 6. **Answer part b:** When will Francis have more than $1000$ dollars? Set up inequality: $$43x + 27 > 1000$$ 7. **Solve the inequality:** $$43x + 27 > 1000$$ $$43x > 1000 - 27$$ $$43x > 973$$ $$x > \frac{973}{43}$$ 8. **Simplify the fraction:** $$x > \frac{\cancel{973}}{\cancel{43}}$$ Since $43 \times 22 = 946$ and $43 \times 23 = 989$, $x$ is approximately $22.63$. 9. **Interpret the result:** Francis will have more than $1000$ dollars after about $23$ days (since $x$ must be a whole number of days). **Final answers:** - a. $y = 43x + 27$ - b. Francis will have more than $1000$ dollars after $23$ days.