Systems of Equations and Applications – Step-by-Step Study Guidance

Q1. The cost of 5 squash and 2 zucchini is $1.32. Three squash and 1 zucchini cost $0.75. Find the cost of each vegetable.

Background

Topic: Systems of Linear Equations (Standard Form)

This question tests your ability to set up and solve a system of equations to find the cost of two different items given two total cost scenarios.

Key Terms and Formulas:

• Let $x$ = cost of one squash

• Let $y$ = cost of one zucchini

• Standard form of a linear equation: $Ax + By = C$

Step-by-Step Guidance

1. Assign variables: Let $x$ be the cost of one squash and $y$ be the cost of one zucchini.

2. Write the first equation based on the first sentence: $5x + 2y = 1.32$

3. Write the second equation based on the second sentence: $3x + y = 0.75$

4. Set up the system of equations:

   $\begin{cases} 5x + 2y = 1.32 \\ 3x + y = 0.75 \end{cases}$

5. Decide on a method to solve (substitution or elimination). For example, you could solve the second equation for $y$ and substitute into the first equation.

Try solving on your own before revealing the answer!

Q2. Judy worked 8 hours and Ben worked 10 hours. Their combined pay was $80. When Judy worked 9 hours and Ben worked 5 hours, their combined pay was $65. Find the hourly rate of pay for each person.

Background

Topic: Systems of Linear Equations (Word Problems)

This question asks you to set up and solve a system of equations to find two unknown hourly rates given two different work/pay scenarios.

Key Terms and Formulas:

• Let $x$ = Judy's hourly rate

• Let $y$ = Ben's hourly rate

• Equation format: (hours worked) × (hourly rate) = total pay

Step-by-Step Guidance

1. Assign variables: Let $x$ be Judy's hourly rate and $y$ be Ben's hourly rate.

2. Write the first equation: $8x + 10y = 80$

3. Write the second equation: $9x + 5y = 65$

4. Set up the system:

   $\begin{cases} 8x + 10y = 80 \\ 9x + 5y = 65 \end{cases}$

5. Choose a method (elimination or substitution) to solve for one variable first.

Try solving on your own before revealing the answer!

Q3. Rob has 40 coins, all dimes and quarters, worth $7.60. How many dimes and how many quarters does he have?

Background

Topic: Systems of Linear Equations (Coin Problems)

This question tests your ability to set up and solve a system of equations involving the number and value of coins.

Key Terms and Formulas:

• Let $x$ = number of dimes

• Let $y$ = number of quarters

• Value of a dime = $0.10$

• Value of a quarter = $0.25$

Step-by-Step Guidance

1. Assign variables: $x$ for dimes, $y$ for quarters.

2. Write the first equation for the total number of coins: $x + y = 40$

3. Write the second equation for the total value: $0.10x + 0.25y = 7.60$

4. Set up the system:

   $\begin{cases} x + y = 40 \\ 0.10x + 0.25y = 7.60 \end{cases}$

5. Solve one equation for one variable and substitute into the other equation.

Try solving on your own before revealing the answer!

Q4. Kelly has 24 dimes and quarters worth $3.60. How many quarters does she have?

Background

Topic: Systems of Linear Equations (Coin Problems)

This question asks you to set up and solve a system of equations involving the number and value of coins, with a focus on finding the number of quarters.

Key Terms and Formulas:

• Let $x$ = number of dimes

• Let $y$ = number of quarters

• Value of a dime = $0.10$

• Value of a quarter = $0.25$

Step-by-Step Guidance

1. Assign variables: $x$ for dimes, $y$ for quarters.

2. Write the first equation for the total number of coins: $x + y = 24$

3. Write the second equation for the total value: $0.10x + 0.25y = 3.60$

4. Set up the system:

   $\begin{cases} x + y = 24 \\ 0.10x + 0.25y = 3.60 \end{cases}$

5. Solve for $y$ (number of quarters) using substitution or elimination.

Try solving on your own before revealing the answer!

Q5. The talent show committee sold a total of 530 tickets in advance. Student tickets cost $3 each and the adult tickets cost $4 each. If the total receipts were $1740, how many of each type of ticket were sold?

Background

Topic: Systems of Linear Equations (Ticket Sales)

This question tests your ability to set up and solve a system of equations involving totals and prices.

Key Terms and Formulas:

• Let $x$ = number of student tickets

• Let $y$ = number of adult tickets

• Student ticket price = $3$

• Adult ticket price = $4$

Step-by-Step Guidance

1. Assign variables: $x$ for student tickets, $y$ for adult tickets.

2. Write the first equation for the total number of tickets: $x + y = 530$

3. Write the second equation for the total receipts: $3x + 4y = 1740$

4. Set up the system:

   $\begin{cases} x + y = 530 \\ 3x + 4y = 1740 \end{cases}$

5. Solve for one variable and substitute into the other equation.

Try solving on your own before revealing the answer!