Study guide: Plan with Linear Rules
Evaluate linear models, recover them from tables, find inputs, and compare intersecting rules.
Ideas to explain in your own words
- Interpret the fixed part and changing part of a linear function.
- Calculate rate over a multi-unit interval and recover the fixed part.
- Solve a linear rule for a specified output and check its domain.
- Solve an intersection and interpret costs on either side.
Watch your reasoning
- Substitute x = 4 into the second rule: 3 × 4 + 1 = $13. The coefficient is a per-poster rate, and the 1 is added once.
- The rate is (16 − 7) ÷ (4 − 1) = 9 ÷ 3 = 3. The fixed part is 7 − 3 × 1 = 4, so y = 3x + 4. It also gives 25 at x = 7.
- Subtract the five-dollar fee: 33 − 5 = 28. Divide by four: x = 7. Checking 4 × 7 + 5 = 33 confirms the order.
- 2x + 9 = 3x + 3 gives 9 = x + 3, so x = 6. Both costs are $21: 12 + 9 = 18 + 3.
A worked review example
Connect a rule, table and rate: For x = 1, 2, 3, the totals are $5, $7, $9. For 4 posters, y = 2 × 4 + 3 = $11. A zero-poster input gives $3 in the mathematical rule; whether a real booth accepts such an order is a separate policy. Find a Linear Rule from a Table: The rate is 12 ÷ 3 = 4. The fixed amount is 11 − 4 × 2 = 3. Thus y = 4x + 3, and at x = 8 it gives 32 + 3 = 35. Work Backward Through a Linear Rule: 4x + 5 = 25 gives 4x = 20, so x = 5 posters. The check is 4 × 5 + 5 = 25 dollars. Find When Two Linear Rules Agree: At x = 4, A = 8 + 6 = 14 and B = 12 + 2 = 14. At x = 2, A = 10 and B = 8. At x = 6, A = 18 and B = 20.
Make a practice plan
- Try the supplied lesson problems without looking at the answers.
- Compare your method with the worked examples and explain any correction.
- Use the new review cases, then complete the transfer task on paper or in your own document.