Question

math connection - writing and solving linear equations
for each scenario, write a linear equation to model what’s happening, and then solve the equation to answer the question.

1. melody has $50 in her savings account right now. she wants to save $250 for a taylor swift concert ticket. she figures she can save $20/week from her part - time job.

a. write an equation to model her savings account balance after x weeks.
b. how long will it take to be able to afford the ticket?

1. after a really rough winter, kristoff cant afford to fix his sled until he has $7000 saved. every month, his take - home pay is $2500, and he uses the 50/30/20 rule to determine how much hell spend and save.

a. write an equation to model kristofs savings after x months.
b. how long will it be before he can get his sled fixed if hes already got $900 in the bank?

1. raven is starting with absolutely no savings, but she wants to have an emergency fund of $10,000 saved by the end of her first year at her new job.

a. how much will she need to save per month in order to meet this savings goal?
b. what would ravens take - home pay need to be in order for that monthly savings target to represent 20% of her pay?

1. fill - in - the blanks to write a word problem for your own personal savings goal.

right now i have ____ already saved. this money is located in ________ i would like to have a total of __ in order to _______________________

1. select one of these two statements to complete:

☐ i can save ____ per month, which means it will take me ____ to reach my goal.
☐ i would like to reach my goal in ____ months, so ill need to save ____ per month.

Explanation:

Problem 1:

Part a

Step1: Define variables & equation

Let $y$ = total savings. Initial savings: $\$50$, weekly save: $\$20$.
$y = 50 + 20x$

Part b

Step1: Set target savings

Set $y=250$, solve for $x$.
$250 = 50 + 20x$

Step2: Isolate variable term

Subtract 50 from both sides.
$250 - 50 = 20x$
$200 = 20x$

Step3: Solve for x

Divide by 20.
$x = \frac{200}{20} = 10$

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Problem 2:

Part a

Step1: Calculate monthly savings

20% of $\$2500$ is saved.
$\text{Monthly savings} = 0.20 \times 2500 = 500$

Step2: Write savings equation

Let $y$ = total savings.
$y = 500x$

Part b

Step1: Set target with initial savings

Target: $\$7000$, initial: $\$900$.
$7000 = 900 + 500x$

Step2: Isolate variable term

Subtract 900 from both sides.
$7000 - 900 = 500x$
$6100 = 500x$

Step3: Solve for x

Divide by 500.
$x = \frac{6100}{500} = 12.2$

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Problem 3:

Part a

Step1: Define time & target

1 year = 12 months, target: $\$10,000$.
Let $m$ = monthly savings.
$12m = 10000$

Step2: Solve for m

Divide by 12.
$m = \frac{10000}{12} \approx 833.33$

Part b

Step1: Set up pay equation

Let $p$ = take-home pay, 20% of $p$ = monthly savings.
$0.20p = \frac{10000}{12}$

Step2: Solve for p

Divide by 0.20.
$p = \frac{10000}{12 \times 0.20} = \frac{10000}{2.4} \approx 4166.67$

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Problem 4 (Example):

(Note: This is a personalization question; example provided)
Right now I have $\$1000$ already saved. This money is located in a high-yield savings account. I would like to have a total of $\$5000$ in order to buy a used car.

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Problem 5 (Example matching Problem 4):

Select one option:
$\square$ I can save $\$333.33$ per month, which means it will take me 12 months to reach my goal.
(or)
$\square$ I would like to reach my goal in 12 months, so I'll need to save $\$333.33$ per month.

Answer:

1. a. $y = 50 + 20x$

b. 10 weeks

1. a. $y = 500x$

b. 12.2 months

1. a. $\approx\$833.33$ per month

b. $\approx\$4166.67$

1. (Example) Right now I have $\$1000$ already saved. This money is located in a high-yield savings account. I would like to have a total of $\$5000$ in order to buy a used car.
2. (Example matching 4) $\square$ I can save $\$333.33$ per month, which means it will take me 12 months to reach my goal.