1. **State the problem:** We need to estimate from the graph how many rocking chairs the carpenter must sell to break even, then write and solve a system of equations to verify the estimate. 2. **Understanding break-even:** Break-even occurs when income equals expense. So, we find the point where the income and expense lines intersect. 3. **Estimate from the graph:** The income points given are (50, 670) and (75, 420), and the expense point is (0, 800). Both lines slope downward. 4. **Find equations of the lines:** - For income line through points (50, 670) and (75, 420): $$m = \frac{420 - 670}{75 - 50} = \frac{-250}{25} = -10$$ Equation form: $$y = mx + b$$ Using point (50, 670): $$670 = -10 \times 50 + b \Rightarrow b = 670 + 500 = 1170$$ So, income line: $$y = -10x + 1170$$ - For expense line through points (0, 800) and approximate slope (since only one point given, assume expense decreases linearly): We need another point; assume expense decreases similarly. Since no other point is given, let's assume expense is constant at 800 (horizontal line): $$y = 800$$ 5. **Solve system:** Set income equal to expense: $$-10x + 1170 = 800$$ $$-10x = 800 - 1170 = -370$$ $$x = \frac{370}{10} = 37$$ 6. **Interpretation:** The carpenter must sell approximately 37 chairs to break even. 7. **Check:** Income at 37 chairs: $$y = -10(37) + 1170 = -370 + 1170 = 800$$ Matches expense, confirming break-even. **Final answer:** The carpenter must sell about 37 rocking chairs to break even.