Lesson 9
When Are They the Same?
Problem 1
Cell phone Plan A costs $70 per month and comes with a free $500 phone. Cell phone Plan B costs $50 per month but does not come with a phone. If you buy the $500 phone and choose Plan B, how many months is it until your cost is the same as Plan A's?
Solution
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Problem 2
Priya and Han are biking in the same direction on the same path.
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Han is riding at a constant speed of 16 miles per hour. Write an expression that shows how many miles Han has gone after \(t\) hours.
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Priya started riding a half hour before Han. If Han has been riding for \(t\) hours, how long has Priya been riding?
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Priya is riding at a constant speed of 12 miles per hour. Write an expression that shows how many miles Priya has gone after Han has been riding for \(t\) hours.
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Use your expressions to find when Han and Priya meet.
Solution
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Problem 3
Which story matches the equation \(\text-6+3x=2+4x\)?
At 5 p.m., the temperatures recorded at two weather stations in Antarctica are -6 degrees and 2 degrees. The temperature changes at the same constant rate, \(x\) degrees per hour, throughout the night at both locations. The temperature at the first station 3 hours after this recording is the same as the temperature at the second station 4 hours after this recording.
Elena and Kiran play a card game. Every time they collect a pair of matching cards, they earn \(x\) points. At one point in the game, Kiran has -6 points and Elena has 2 points. After Elena collects 3 pairs and Kiran collects 4 pairs, they have the same number of points.
Solution
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Problem 4
For what value of \(x\) do the expressions \(\frac23x+2\) and \(\frac43x-6\) have the same value?
Solution
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Problem 5
Decide whether each equation is true for all, one, or no values of \(x\).
- \(2x+8=\text-3.5x+19\)
- \(9(x-2)=7x+5\)
- \(3(3x+2)-2x=7x+6\)
Solution
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(From Unit 4, Lesson 8.)Problem 6
Solve each equation. Explain your reasoning.
\(3d+16 = \text-2(5-3d)\)
\(2k-3(4-k)=3k+4\)
\(\frac{3y-6}{9}=\frac{4-2y}{\text-3}\)
Solution
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(From Unit 4, Lesson 6.)Problem 7
Describe a rigid transformation that takes Polygon A to Polygon B.
Solution
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(From Unit 1, Lesson 7.)