Zachary purchased a computer for ​$1 comma 8001,800 on a payment plan. FiveFive months after he purchased the​ computer, his balance was ​$1 comma 2251,225. NineNine months after he purchased the​ computer, his balance was ​$765765. What is an equation that models the balance y after x​ months?

Answer 1

To create an equation that models Zachary's balance \( y \) after \( x \) months, we can start by using the information provided to identify a linear relationship between the balance and time.

We have the following data points:

1. At \( x = 5 \) months, \( y = 1225 \)
2. At \( x = 9 \) months, \( y = 765 \)

We can assume a linear equation of the form: \[ y = mx + b \] where \( m \) is the slope and \( b \) is the y-intercept.

Step 1: Calculate the slope \( m \)

The slope \( m \) can be calculated using the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Let's plug in the points:

• \((x_1, y_1) = (5, 1225)\)
• \((x_2, y_2) = (9, 765)\)

Calculating the slope: \[ m = \frac{765 - 1225}{9 - 5} = \frac{-460}{4} = -115 \]

Step 2: Calculate the y-intercept \( b \)

To find \( b \), we can use one of the points. We'll use \( (5, 1225) \): \[ y = mx + b \] Substituting \( m \) and the point: \[ 1225 = -115(5) + b \] Calculating it: \[ 1225 = -575 + b \] \[ b = 1225 + 575 = 1800 \]

Final Equation

The equation that models the balance \( y \) after \( x \) months is: \[ y = -115x + 1800 \]

Conclusion

This linear equation represents Zachary's balance after \( x \) months of payments for his computer.