Work Backward Through a Linear Rule

Supplied materials

• Paper or a note app.
• Fictional print rule: y = 4x + 5 dollars, where x is a nonnegative whole poster count.

Learn the idea

A function rule can tell us an output from an input, but a planning question may give the total cost and ask for the count. Write an equation using that cost, then undo the fixed addition before undoing multiplication.

The algebraic solution must fit the context. Poster counts are whole and nonnegative. A value such as 2.5 can solve an equation but cannot represent an exact order of whole posters under this model. Substitute valid answers into the original rule.

1. Set 4x + 5 = 25 for a twenty-five-dollar order.
2. Subtract five, then divide twenty by four.
3. Check that x = 5 is a whole count and rebuild its cost; solve the new target below.

Check your reasoning

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Take it further

Change the target to $15. Algebra gives x = (15 − 5) ÷ 4 = 2.5. There is no whole-poster order costing exactly $15; two cost $13 and three cost $17.

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