A couple purchased a house in 2005. In 2009 the house appraised for $475,000. The couple refinanced the house in 2014 and the appraisal was $468,000. The couple determined a linear relationship existed between the the number of years since 2005 and the appraised value of the house.
Write a linear equation that models this relationship.
V(t) 475,000+ (475,000-468,800)
In thousands, let t be the number of years since 2009, since that is the year we have our first appraisal.
V(t) = 475+(468-475)/(2014-2009) t
= 475 - 1.4t
But we want t to be years since 2005. That means we have to add back 4 more years of loss: 4*1/4=5.6
V(t) = 480.6 - 1.4t
or, going back to dollars,
V(t) = 480,600 - 1400t
So how did you get 1.4t in this equation
To write a linear equation that models the relationship between the number of years since 2005 and the appraised value of the house, we need to find the slope and the y-intercept. The slope represents the rate at which the appraised value changes per year, and the y-intercept represents the initial value in 2005.
We have the following information:
In 2005, the appraised value is $475,000, which can be represented by (0, 475,000).
In 2009, the appraised value is $468,000, which can be represented by (4, 468,000).
To find the slope, we need to use the slope formula:
slope = (y2 - y1) / (x2 - x1)
Using the data points (0, 475,000) and (4, 468,000), we can substitute the values into the formula:
slope = (468,000 - 475,000) / (4 - 0)
= (-7,000) / 4
= -1,750
So, the slope of the linear equation is -1,750.
To find the y-intercept, we can substitute the values of one of the points (0, 475,000) and the slope into the slope-intercept form of a linear equation:
y = mx + b
where m is the slope and b is the y-intercept.
Using the point (0, 475,000) and the slope -1,750, we can substitute the values into the equation:
475,000 = -1,750(0) + b
b = 475,000
Therefore, the y-intercept of the linear equation is 475,000.
Putting everything together, the linear equation that models the relationship between the years since 2005 (t) and the appraised value (V) is:
V(t) = -1,750t + 475,000