Lucas took out a loan from his family member to buy a car. After 12 months, he had paid $5,140. After 36 months, he paid $13,420. Create a linear equation that represents the amount of money he has paid as a function of months.
To find the linear equation that represents the amount of money Lucas paid as a function of months, we can use the slope-intercept form:
y = mx + b
where y is the amount of money paid, x is the number of months, m is the slope, and b is the y-intercept.
First, we need to find the slope, which represents the rate at which Lucas is repaying the loan. The slope can be calculated by finding the change in the amount of money paid (Δy) divided by the change in the number of months (Δx):
m = (y2 - y1) / (x2 - x1)
Using the two given points (12 months, $5,140) and (36 months, $13,420):
m = ($13,420 - $5,140) / (36 - 12)
m = $8,280 / 24
m = $345
Now that we have the slope, we can substitute it into the slope-intercept form along with one of the given points to find the y-intercept (b).
Using the point (12 months, $5,140):
$5,140 = $345(12) + b
$5,140 = $4,140 + b
b = $5,140 - $4,140
b = $1,000
The linear equation that represents the amount of money Lucas has paid as a function of months is:
y = $345x + $1,000