Zachary purchased a computer for 

​$1 comma 1001,100 

on a payment plan. 

ThreeThree 

months after he purchased the​ computer, his balance was 

​$800800. 

FiveFive 

months after he purchased the​ computer, his balance was 

​$600600. 

What is an equation that models the balance y after x​ months?

To model the balance $y$ after $x$ months, we can assume the balance decreases linearly over time. We need to find the equation of a line that fits the given data.

### Step 1: Identify the points
We have two points that represent the time $x$ in months and the balance $y$:

- After 3 months: $(3, 800)$
- After 5 months: $(5, 600)$

### Step 2: Find the slope $m$
The slope $m$ of the line can be found using the formula:

\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]

Substitute the points $(x_1, y_1) = (3, 800)$ and $(x_2, y_2) = (5, 600)$:

\[
m = \frac{600 - 800}{5 - 3} = \frac{-200}{2} = -100
\]

### Step 3: Use the point-slope form
Now, using the point-slope form $y - y_1 = m(x - x_1)$, we can use one of the points, say $(3, 800)$:

\[
y - 800 = -100(x - 3)
\]

### Step 4: Simplify to slope-intercept form
Simplify the equation to get it into the slope-intercept form $y = mx + b$:

\[
y - 800 = -100x + 300
\]

\[
y = -100x + 1100
\]

### Final Answer
The equation that models the balance $y$ after $x$ months is:

\[
y = -100x + 1100
\]

Would you like more details or have any questions?

Here are some related questions:
1. How do you interpret the slope of this equation in the context of the payment plan?
2. How can you determine the number of months it will take for the balance to reach zero?
3. What would the equation be if the payments were increasing the balance instead of decreasing it?
4. How do you find the y-intercept of a line given two points?
5. What would the equation be if the balance started at $900 instead of $1,100?

**Tip:** When dealing with payment plans or similar financial models, the slope represents the rate of change in balance, which is often the payment amount or decrease per period.

Explore how to model the balance after purchasing a computer using a linear equation. Learn the step-by-step process to find the equation based on given data points. Suitable for Grades 7-9.

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