Sydney borrowed money from a friend and is paying back the...

Sydney borrowed money from a friend and is paying back the loan. The remaining amount she owes, $A$, can be calculated by the equation $A=870-30 w$, where $w$ represents the number of weeks since she took the loan. What does the number 870 represent? A) The total amount of money Sydney has repaid B) The amount of money Sydney repays each week C) The number of weeks Sydney has been paying back the loan D) The original amount that Sydney borrowed

Sydney borrowed money from a friend and is paying back the loan. The remaining amount she owes, $A$, can be calculated by the equation $A=870-30 w$, where $w$ represents the number of weeks since she took the loan. What does the number 870 represent?
A) The total amount of money Sydney has repaid
B) The amount of money Sydney repays each week
C) The number of weeks Sydney has been paying back the loan
D) The original amount that Sydney borrowed

Key Concepts

Y-intercept/Initial Value

In a linear equation of the form y = mx + b, the constant term (b) represents the initial value of the quantity before any changes occur—in this context, it is the original amount borrowed before any repayments are made.

Linear Modeling

Linear equations are used to model relationships between quantities, where one quantity changes at a constant rate relative to another. Understanding this concept is essential for interpreting financial models like loan repayments, where the change is consistent over time.

Transcript

00:01 So we're told that this person has taken out a loan of $8 ,250 and is paying off this loan at a rate of $125 per month.

00:09 And in part a, we're going to find an equation which takes in this variable t, which is the amount of months that's gone by and has given us back the amount that is still left on this loan.

00:20 So this is going to be equal to the amount that the loan is equal to, so 8 ,250 minus the amount that we are paying each month.

00:29 So we're paying $125 per month, and then we need to multiply this by the amount of months to then give us the total amount of money that we've paid after t months.

00:39 So this is our equation for the amount of money left in our loan.

00:43 In part b, we're going to figure out at what time or after how many months will our p of t or the amount left in our loan be equal to 5 ,000.

00:53 So we're going to set p of t equal to 5 ,000.

00:57 So 5 ,000 is equal to 8 ,000.

00:59 250 minus 125 t and i'm going to plus this 125t over and i'm also going to minus this 5 000 over to this side so 5 or 8 ,250 minus 5 000 is 3 ,250 so we get 125 t is equal to 3 ,250 and now we can just divide by 125 on both sides to find t and t and t and this case is equal to 26 months.

01:32 So it will take 26 months for this loan to be down to $5 ,000.

01:40 Now for part c, what we're going to do is we're actually going to graph our equation p of t.

01:47 So p of t is equal to 8 ,250 minus 125t.

01:55 So what we're going to do is actually graph this.

01:58 So i'm going to figure out what the, or how many months it'll take to actually get to get this loan down to zero.

02:08 So i'm going to set p of t equal to zero, 8 ,250 minus 125t.

02:15 We can add 125t over.

02:21 So you get 125t is equal to 8 ,250 and t, in this case is equal to 66.

02:29 Months...