Question

Sarah is saving up to buy a new snowboard and boots for the winter that costs $850. She has already saved some money and babysits each week. After 2 weeks, she has saved up $305 and after 5 weeks she has $560. What linear equation models this relationship?

Answer 1

To find the linear equation that models Sarah's savings over time, we can first calculate the amount she is saving each week.

From the information given:

- After 2 weeks, she has $305.
- After 5 weeks, she has $560.

Using this information, we can find the increase in savings from week 2 to week 5:

1. Savings after 2 weeks: \( S(2) = 305 \)
2. Savings after 5 weeks: \( S(5) = 560 \)

Now, let's find the change in savings over 3 weeks (from week 2 to week 5):

\[
\Delta S = S(5) - S(2) = 560 - 305 = 255
\]

Now, we divide this difference by the number of weeks to find the weekly savings rate:

\[
\text{Weekly savings rate} = \frac{\Delta S}{\Delta t} = \frac{255}{3} = 85
\]

So, Sarah is saving $85 per week.

Next, to find the y-intercept (the amount she had saved before she started saving weekly), we can use the savings after 2 weeks:

Let \( y \) be the total savings and \( x \) be the number of weeks. The general linear equation can be expressed as:

\[
y = mx + b
\]

Where \( m \) is the slope (weekly savings) and \( b \) is the y-intercept. We found \( m = 85 \).

Now we can use the point (2, 305) to find \( b \):

\[
305 = 85(2) + b
\]

Calculating this gives:

\[
305 = 170 + b
\]

Subtracting 170 from both sides:

\[
b = 305 - 170 = 135
\]

Thus, the equation modeling Sarah's savings is:

\[
y = 85x + 135
\]

This linear equation indicates that Sarah's total savings is $135 initially (before she starts babysitting) plus $85 for each week she babysits.