# Lesson 9

When Are They the Same?

Let’s use equations to think about situations.

### 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?

### Problem 2

Priya and Han are biking in the same direction on the same path.

1. 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.

2. Priya started riding a half hour before Han. If Han has been riding for $$t$$ hours, how long has Priya been riding?

3. 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.

4. Use your expressions to find when Han and Priya meet.

### Problem 3

Which story matches the equation $$\text-6+3x=2+4x$$?

A:

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.

B:

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.

### Problem 4

For what value of $$x$$ do the expressions $$\frac23x+2$$ and $$\frac43x-6$$ have the same value?

### Problem 5

Decide whether each equation is true for all, one, or no values of $$x$$.

1. $$2x+8=\text-3.5x+19$$
2. $$9(x-2)=7x+5$$
3. $$3(3x+2)-2x=7x+6$$
(From Unit 4, Lesson 8.)

### Problem 6

Solve each equation. Explain your reasoning.

$$3d+16 = \text-2(5-3d)$$

$$2k-3(4-k)=3k+4$$

$$\frac{3y-6}{9}=\frac{4-2y}{\text-3}$$

(From Unit 4, Lesson 6.)

### Problem 7

Describe a rigid transformation that takes Polygon A to Polygon B.