Find When Two Linear Rules Agree

Supplied materials

• Paper or a note app.
• Fictional print prices: A = 2x + 6 and B = 3x + 2 dollars; x is a nonnegative whole poster count.

Learn the idea

Two rules agree at an input where both produce the same output. Set their expressions equal: 2x + 6 = 3x + 2. This equation compares complete costs, including both fixed fees and changing parts.

Subtract 2x from both sides and subtract two to find x = 4. The common output is fourteen dollars. Because B rises faster, B is lower before the crossing and higher after it; checking one count on each side makes that interpretation concrete.

1. Set the two complete cost expressions equal.
2. Solve 2x + 6 = 3x + 2, then substitute x = 4 into both rules.
3. Compare both costs at x = 2 and x = 6, then solve the changed pair below.

Check your reasoning

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Take it further

Change D to 3x + 5 while C stays 2x + 9. Equality now gives x = 4 and a shared cost of $17. Raising D’s fixed fee moves the crossing to a smaller count.

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