Question

mixing real - world and mathematical problems related to systems of equations
mild\tmedium\tspicy
student can write a system of equations to model real - world problems\tstudent can write a system of equations to model real - world problems and then solve the system. they can interpret and check their solutions.\tstudent can write a system of equations to model real - world problems, solve the system, and interpret the solution. they can also analyze when the system has no solution or infinitely many solutions.

1. pizza parlor sells a large pizza for $9 plus $1.25 for each additional topping. prime pizza sells a large pizza for $7.50 plus $1.75 for each additional topping. write a system of equations that could be used to determine for which number of toppings both pizzas are the same price.
2. sarah has $350 in her bank account. she spends $8.50 each week for a gaming subscription and saves the rest of her money.

aaron has $135 in his bank account. he gets a tutoring job and earns $24.50 per week.
after how many weeks will sarah and aaron have the same amount of money in their account? what will that amount be? write and solve a system of equations. show all work.

1. greg has $2.30, made up of only dimes and nickels. he has 29 coins in total. how many dimes and how many nickels does greg have? write and solve a system of equations. show all work.

Explanation:

First Problem (Pizza Prices)

Step1: Define variables

Let \( x \) be the number of toppings and \( y \) be the total price of a large pizza.

Step2: Write equation for Pizza Parlor

Pizza Parlor: Base price is $9, plus $1.25 per topping. So \( y = 9 + 1.25x \)

Step3: Write equation for Prime Pizza

Prime Pizza: Base price is $7.50, plus $1.75 per topping. So \( y = 7.50 + 1.75x \)

Step1: Define variables

Let \( w \) be the number of weeks and \( m \) be the amount of money in the account.

Step2: Write equation for Sarah

Sarah starts with $350 and spends $8.50 per week. So \( m = 350 - 8.50w \)

Step3: Write equation for Aaron

Aaron starts with $135 and earns $24.50 per week. So \( m = 135 + 24.50w \)

Step4: Set equations equal to solve for \( w \)

\( 350 - 8.50w = 135 + 24.50w \)

Step5: Solve for \( w \)

Add \( 8.50w \) to both sides: \( 350 = 135 + 33w \)
Subtract 135 from both sides: \( 215 = 33w \)
\( w=\frac{215}{33}\approx 6.52 \) (weeks)

Step6: Find \( m \)

Substitute \( w\approx 6.52 \) into Sarah's equation: \( m = 350 - 8.50\times6.52\approx 350 - 55.42 = 294.58 \)
Or into Aaron's equation: \( m = 135 + 24.50\times6.52\approx 135 + 159.74 = 294.74 \) (minor difference due to rounding \( w \))
To get an exact value, solve \( 350 - 8.5w = 135 + 24.5w \)
\( 350 - 135 = 24.5w + 8.5w \)
\( 215 = 33w \)
\( w=\frac{215}{33}=\frac{215\div1}{33\div1}=\frac{215}{33}\approx 6.52 \) weeks. Then \( m = 350 - 8.5\times\frac{215}{33}=350-\frac{1827.5}{33}=\frac{11550 - 1827.5}{33}=\frac{9722.5}{33}\approx 294.62 \)

Step1: Define variables

Let \( d \) be the number of dimes and \( n \) be the number of nickels.

Step2: Write equation for total coins

Total coins: \( d + n = 28 \)

Step3: Write equation for total value

A dime is $0.10, a nickel is $0.05, total value is $2.30. So \( 0.10d + 0.05n = 2.30 \)

Step4: Solve the system

From the first equation, \( d = 28 - n \)
Substitute into the second equation: \( 0.10(28 - n) + 0.05n = 2.30 \)

Step5: Simplify and solve for \( n \)

\( 2.8 - 0.10n + 0.05n = 2.30 \)
\( 2.8 - 0.05n = 2.30 \)
Subtract 2.8 from both sides: \( -0.05n = -0.5 \)
Divide by -0.05: \( n = 10 \)

Step6: Find \( d \)

\( d = 28 - n = 28 - 10 = 18 \)

Answer:

The system of equations is \(

$$\begin{cases} y = 9 + 1.25x \\ y = 7.50 + 1.75x \end{cases}$$

\)

Second Problem (Sarah and Aaron's Money)