Let's tackle the two problems shown in the image step by step.

1. Rectangle Perimeter Problem (Problem 9):

The problem asks for the value of $x$, given that the perimeter of the rectangle is 65 inches. The dimensions of the rectangle involve fractions and algebraic expressions.

The perimeter of a rectangle is calculated as: $P = 2 \times (\text{Length} + \text{Width})$

In the problem:

• Length = $\frac{13}{4} + 2x$
• Width = $3x + 1$

The perimeter is given as 65 inches, so the equation becomes: $65 = 2 \times \left( \left( \frac{13}{4} + 2x \right) + \left( 3x + 1 \right) \right)$

Now, solving step by step: $65 = 2 \times \left( \frac{13}{4} + 2x + 3x + 1 \right)$ $65 = 2 \times \left( \frac{13}{4} + 5x + 1 \right)$ $65 = 2 \times \left( \frac{13}{4} + \frac{4}{4} + 5x \right)$ $65 = 2 \times \left( \frac{17}{4} + 5x \right)$ Now, distribute the 2 on both terms: $65 = \frac{34}{4} + 10x$ Convert $\frac{34}{4}$ into a decimal: $65 = 8.5 + 10x$ Subtract 8.5 from both sides: $65 - 8.5 = 10x$ $56.5 = 10x$ Now, solve for $x$: $x = \frac{56.5}{10} = 5.65$

So, the value of $x$ is approximately 5.65.

2. Meal Price Problem (Problem 10):

James bought a cheeseburger, fries, and a drink. The cheeseburger was three times the price of the fries, and the drink and fries were the same price. The total cost of the meal was $12.50. We need to find the price of each item.

Let:

• The price of the fries be $f$
• The price of the cheeseburger be $3f$ (since it's three times the fries)
• The price of the drink also be $f$ (since it's the same price as the fries)

The total cost is given as: $f + 3f + f = 12.50$ Simplifying: $5f = 12.50$ Now, solve for $f$: $f = \frac{12.50}{5} = 2.50$

Thus:

• Fries cost $f = 2.50$
• Cheeseburger costs $3f = 7.50$
• Drink costs $f = 2.50$

So, the prices are:

• Cheeseburger: $7.50
• Fries: $2.50
• Drink: $2.50

Do you have any questions or would you like more details?

Here are some related questions to explore:

1. How would the problem change if the perimeter of the rectangle were different?
2. Can you derive an expression for the area of the rectangle given the same dimensions?
3. What if James had a discount coupon of 10% for the total meal cost? What would be the new prices?
4. How would the system of equations look if the drink had been half the price of the fries?
5. What other methods can be used to solve the system of equations for James's meal?

Tip: When working with fractions and decimals in algebraic equations, it's often easier to convert everything to decimals early on to simplify calculations.