Linear Equations Practice Problems

These linear equations practice problems replace generic x and y values with checking account balances, weekly paychecks, ATM fees, and check-cashing charges. The goal is to make slope and initial value mean something concrete: slope is a rate of change students already encounter, like a fee per transaction or an amount earned per week, and initial value is a starting balance or a flat fee before that rate kicks in.

The set of eight problems moves through several formats rather than repeating the same question with new numbers. Students identify slope and initial value from equations already written out, build their own equation from a weekly-earnings scenario, match four written scenarios to four equations, and then write and solve equations for account balances that shrink over time because of recurring fees. The last few problems get more specific about real financial products: a flat check-cashing fee, a bank's tiered ATM fee structure, and a side-by-side comparison of two coin-counting machines that charge fees differently, one as a flat rate plus a percentage, and one as a straight percentage.

That last comparison, between the two coin-counting machines, is the strongest problem in the set for connecting the math to something students may have actually used. It asks them to write both fee equations, evaluate them at a specific coin value, and then reason about which machine is the better deal and why, which pushes past just solving for y into interpreting what the slope and initial value mean for a real cost.

Below is the worksheet exactly as it appears in the Rapunzl curriculum, followed by teacher notes on how to use it in class.

This activity is from Module 14 of the Rapunzl curriculum, Financial Equations.

Understanding Linear Equations

Through this activity, students will explore linear equations by examining common financial instruments and investments, such as stocks, bonds, and mutual funds. By understanding how linear equations apply to different financial instruments, students will be able to make informed decisions about their investments and understand the impact of different financial decisions.

1. Identify the slope and initial value for each equation below
   • a. y = -5x + 12
   • b. y = -35x + 120

c.

1. Ahmad has $112 in his checking account and earns $75 per week working part-time. Assume he has no expenses currently.

b. Slope:

c. Write an equation to show his account balance, y, after x weeks.

d. How much does he have after one year (52 weeks)? Use the equation you wrote in part c.

1. Match each equation with the situation it represents, where y is account balance and x is number of weeks.

1. Fatima has $207 in her account. A $7.25 fee is charged each month the balance is below $100. She withdraws $120 one time.
   • a. Write an equation to model her balance after x months since the withdrawal, if she makes no further deposits or withdrawals.
   • b. Solve for y when x=4. What does that mean in this context?
   • c. After how many months will Fatima’s balance reach $0 or below?
2. Leyla’s old account had a $3.50 ATM fee, so she decided to change to a free student checking account with no ATM fees.
   • a. Write an equation to represent how much money Leyla saves on fees (y) depending on how many times she uses an ATM (x).
   • b. How much does Leyla save if she uses the ATM 8 times?
3. Walmart charges a $4 fee to cash a check that is worth up to $1000.
   • a. Write an equation to represent the total amount you’ll pay in fees, y, if you deposit x number of $500 checks.
   • b. Imagine you deposit four checks per month. Write an equation to model the total you pay in check cashing fees, y, after x months.
4. Many grocery stores have machines that will give cash for your change but charge a fee.
   • a. Coinstar charges an average fee of 11.9% of your coins’ value. Write an equation that models the total fee you pay (y) based on the value of your coins (x) if you use a Coinstar machine.
   • b. ChangeMaker has a coin machine that charges a $1 flat fee, plus 5% of your coins’ value. Write an equation to model the fee you pay (y) based on the value of your coins (x).
   • c. Imagine you have $13.65 in coins that you want to exchange for bills. How much would you pay in fees at each machine? Round your answer to the nearest cent.
   • d. When is the ChangeMaker a better deal than the Coinstar? Why?
5. Angel’s bank allows 3 free ATM withdrawals each month, then charges a $5 ATM fee.
   • a. Write an equation to model how much Angel will pay in fees (y) based on their number of ATM withdrawals (x).
   • b. Based on your equation, how much will Angel pay in fees if they make 0 withdrawals? Why doesn’t that value make sense in the context of the problem?

Teacher Notes

Question 1c has no equation printed with it. If your class is working from the accompanying slide or board example for this activity, use that version; otherwise, skip it or substitute an equation of your own before assigning the set.

Question 2's table sets up initial value and slope before asking students to write the full equation in part c. Have students fill in that table first rather than jumping straight to the equation. It catches a common error early: students who reverse which number is the rate and which is the starting balance.

Question 3's matching exercise is a fast way to check whether students can read the sign of the slope and the sign of the initial value directly out of a written scenario, without doing any calculation. It works well as a quick formative check before question 4, which asks students to build a similar equation from scratch and then solve it.

Question 7 is worth protecting time for. Comparing Coinstar's percentage-only fee to ChangeMaker's flat-fee-plus-percentage structure is where slope and initial value stop being abstract and start explaining an actual price difference. Let students work out both equations before revealing which machine is cheaper at $13.65, so the comparison in part d comes from their own numbers.

This worksheet is one piece of the full Financial Equations unit inside the Rapunzl teacher portal, where activities like this one sit alongside articles, guided practice, and a classroom investing simulator built for grades 6–12.

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