# Lesson 9

Same Situation, Different Symbols

- Let’s think about the how and why of solving equations, and use those ideas to make problems easier.

### 9.1: Math Talk: True Values

For each equation, mentally find the value that makes it true.

- \(25+3 = 21 + x\)
- \(5 \boldcdot 3+15=x \boldcdot 5 +10\)
- \(2- x+8= 2-7 +10\)
- \(2 \boldcdot 12 - 50=3\boldcdot 12 - x\)

### 9.2: Fizzy Drinks and Fast Driving

- Sparkling water and grape juice are mixed together to make 36 ounces of fizzy juice.
- How much sparkling water was used if the mixture contains 19 ounces of grape juice?
- How much grape juice was used if the mixture contains 15 ounces of sparkling water?
- Han wrote the equation, \(x + y = 36\), with \(x\) representing the amount of grape juice used, in ounces, and \(y\) representing the amount of sparkling water used, in ounces. Explain why Han’s equation matches the story.
- Clare wrote the equation \(y = 36 - x\), with \(x\) representing the amount of grape juice used, in ounces, and \(y\) representing the amount of sparkling water used, in ounces. Explain why Clare’s equation matches the story.
- Kiran wrote the equation \(x = y + 36\), with \(x\) representing the amount of grape juice used, in ounces, and \(y\) representing the amount of sparkling water used, in ounces. Explain why Kiran’s equation does
*not*match the story.

- A car is going 65 miles per hour down the highway.
- How far does it travel in 1.5 hours?
- How long does it take the car to travel 130 miles?
- Mai wrote the equation \(y = 65x\), with \(x\) representing the time traveled, in hours, and \(y\) representing the distance traveled, in miles. Explain why Mai’s equation matches the story.
- Tyler wrote the equation \(x = \frac{y}{65}\), with \(x\) representing the time traveled, in hours, and \(y\) representing the distance traveled, in miles. Explain why Tyler’s equation matches the story.
- Lin wrote the equation \(y = \frac{x}{65}\), with \(x\) representing the time traveled, in hours, and \(y\) representing the distance traveled, in miles. Explain why Lin’s equation does
*not*match the story.

### 9.3: Finding an Error

- Situation: The yogurt at Sweet Delights costs \$0.65 per pound and \$0.10 per topping. The total cost of a purchase was \$1.70. Let \(p\) be the weight of the yogurt in pounds and \(t\) be the number of toppings bought.

Tyler’s first and correct equation: \(0.65p+0.10t=1.70\)

Tyler’s second and*incorrect*equation: \(t= (1.70 - .65p) \boldcdot 0.10\)- What is the error?
- What is a correct second equation Tyler could have written?
- What might Tyler have been thinking that led to his mistake?

- Situation: The perimeter of a rectangle (twice the sum of the length and width) is 13.5 inches. Let
*\(l\)*be the length of the rectangle and*\(w\)*be the width of the rectangle.

Tyler’s first and correct equation: \(2(l + w) = 13.5\)

Tyler’s second and*incorrect*equation: \(w = 13.5 - 2l\)- What is the error?
- What is a correct second equation Tyler could have written?
- What might Tyler have been thinking that led to his mistake?

- Situation: For a fundraiser, a school is selling flavored waters for \$2.00 each and pretzels for \$1.50 each. The school has a fundraising goal of \$200. Let \(w\) be the number of waters sold and \(p\) be the number of pretzels sold.

Tyler’s first and correct equation: \(2w + 1.5p = 200\)

Tyler’s second and*incorrect*equation: \(1.5p = 198w\)- What is the error?
- What is a correct second equation Tyler could have written?
- What might Tyler have been thinking that led to his mistake?